The 23rd Thailand-Japan Conference
on Discrete and Computational
Geometry, Graphs, and Games
September 3–5, 2021

Organized by

Department of Mathematics
Faculty of Science, Chiang Mai University
Chiang Mai, Thailand.

The 23rd Thailand-Japan Conference on Discrete and Computational
Geometry, Graphs, and Games (TJCDCG3 2020+1) Book of Abstracts
Organized by Department of Mathematic, Faculty of Science,
Chiang Mai University, Chiang Mai, Thailand.

Table of contents

Front cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iTable of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Abstract (Invited Speakers)
Family Tree of Impossible Objects Created by Optical Illusion
Kokichi Sugihara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Towards Dynamic Voronoi Diagrams
Stefan Langerman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8Understanding the Complexity of Motion Planning through Gadgets
Erik D. Demaine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9Decomposing complete multigraphs into stars of varying sizes
Daniel Horsley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Crossing Parallels
János Pach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Approximation schemes for geometric NP-hard problems: geometry meets algorithms
Jittat Fakcharoenphol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Abstract (Contributed Talks)
A k-ary Middle Levels Conjecture
Xi Shen and Aaron Williams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Every Tree is a Subtree of a Graceful Unicyclic Graph
Sethuraman Guruswamy and Murugan Varadhan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Graph of uv-paths in 2-connected graphs
Eduardo Rivera-Campo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Prime Labeling of Trees Using Eisenstein Integers
Korina Ernjulie Manaloto and Rovin Santos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Unfolding Orthotubes with a Dual Hamiltonian Path
Erik Demaine and Kritkorn Karntikoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Complexity of Simple Folding Orthogonal Crease Patterns
Josh Brunner, Erik D. Demaine, Dylan Hendrickson, Victor Luo and Andy Tockman. . . . . . . . . . . 26

Multifold tiles of polyominoes and convex lattice polygons
Kota Chida, Erik Demaine, Martin Demaine, David Eppstein, Adam Hesterberg, Takashi Horiyama, John
Iacono, Hiro Ito, Stefan Langerman, Ryuhei Uehara and Yushi Uno . . . . . . . . . . . . . . . . . . . . . 28
Orthogonal Fold and Cut
Joshua Ani, Josh Brunner, Erik D. Demaine, Martin L. Demaine, Dylan Hendrickson, Victor Luo andRachana Madhukara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
iv

Redundancy of Linear Codes with Graph Constraints
Ghurumuruhan Ganesan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Random Derangement with Fixed Number of Cycles
Nattawut Phetmak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Multiple Hook Removing Game
Tomoaki Abuku and Masato Tada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Complete Geometric Graphs with no Partition into Plane Spanning Trees
Johannes Obenaus and Joachim Orthaber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
On the Size of Minimal Separators for Treedepth Decomposition
Zijian Xu and Vorapong Suppakitpaisarn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Hamiltonian decompositions of complete 4-partite 3-uniform hypergraphs
Chutima Saengchampa and Chariya Uiyyasathian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
On the Edge Irregularity Strength of Some Disjoint Union Graphs
Rismawati Ramdani, Irsa Islammeidini Rusvianti and Hasni Rahmani Rohim . . . . . . . . . . . . . . . . 44
Tiling the Plane Connectively with Wang Tiles
Chao Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Generating Frames via Discretized Substitution Tilings
Luis Jr. Silvestre and Job Nable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Tiling of the Sphere by Congruent Polygons
Min Yan, Yohji Akama, Hoi Ping Luk and Erxiao Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Renzuru Tilings with Asymmetric Quadrilaterals
Takashi Yoshino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Moving gold sand gameIlya Bogdanov, Grigory Chelnokov and Margarita Akhmejanova . . . . . . . . . . . . . . . . . . . . . . . . 54
Multi-Player Open-Hand BABANUKI
Hironori Kiya and Hirotaka Ono . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Playing impartial games on a simplicial complex as extension of emperor sum
Koki Suetsugu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Losing Positions of Splythoff and 2-Splythoff encoded in the Tribonacci word
Gerard Francis Ortega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Linear-Time Rectilinear Drawings of Triconnected Subcubic Planar Graphs with Orthogonally Convex
Faces
Md. Manzurul Hasan, Debajyoti Mondal and Md. Saidur Rahman . . . . . . . . . . . . . . . . . . . . . . 62
A complete combinatorial characterization of greedy-drawable trees
Reiya Nosaka, Hiroyuki Miyata and Shin-ichi Nakano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Number of Go positions on a connected graphChengyang Qian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
The maximum numbers of the rigid faces and edges in continuous flattening processes of a polyhedron
Kazuki Matsubara and Chie Nara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Minimizing Visible Edges in Polyhedra
Giovanni Viglietta, Csaba Tóth and Jorge Urrutia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70v

Flat folding problem with parallel creases with mountain-valley assignment on a convex polygonal pieceof paperHiro Ito and Sae Neshiba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Max-norm analogs of Euclidean Ramsey theorems
Nóra Frankl, Andrei Kupavskii and Arsenii Sagdeev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Celeste is PSPACE-hard
Lily Chung and Erik Demaine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
NP-Completeness of Peg Solitaire on Graphs
Kazushi Ito and Yasuhiko Takenaga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Prime-Graceful Number
Penying Rochanakul and Sayan Panma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Rainbow Connection Number of Dutch Windmill Graph
Aroonwan Suebsriwichai and Thanasak Mouktonglang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Rainbow Connection Numbers of s-Overlapping r-Uniform Hypertrees
Sitta Alief Farihati, A. N. M Salman and Pritta Etriana Putri . . . . . . . . . . . . . . . . . . . . . . . . 84
Distance to antipode of semi-regular polytope, measured by edges of equal length
Jin Akiyama and Ikuro Sato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Möbius Flowers and Buds
Jin Akiyama, Kiyoko Matsunaga, Sachiko Nakajima and Natsumi Oyamaguchi . . . . . . . . . . . . . . . 88
The Conditions for Fitting a Square in a Trapezoid
Nattapon Rintaew and Supanut Chaidee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Previous Player’s Positions of Impartial Three-Dimensional Chocolate-Bar Games
Ryohei Miyadera and Hikaru Manabe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Chocolate Games and Restricted Nim
Ryohei Miyadera, Hikaru Manabe, Kousei Suzuki, Taishi Aono, Shouei Takahasi and Sohta Kannan . . . 94
Permutation-Generated Maps between Dyck Paths
Kevin Limanta, Hopein Christofen Tang and Yozef Tjandra . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Dynamic Independent Set of Squares
Sujoy Bhore, Jean Cardinal, John Iacono, and Grigorios Koumoutsos . . . . . . . . . . . . . . . . . . . . 98
Γ-Induced-Paired Dominating Graphs of Cycles
Saharath Sanguanpong and Nantapath Trakultraipruk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Biased Domination Game
Tharit Sereekiatdilok and Panupong Vichitkunakorn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Unidirectional monotonic paths through specified points in labeled point sets in convex position
Toshinori Sakai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Logical Matrix Representations in Map Folding
Yiyang Jia, Jun Mitani and Ryuhei Ueharas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Feedback game on Eulerian graphs
Naoki Matsumoto and Atsuki Nagao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Solving teleportation mazes with limited visibility
Robert Barish and Tetsuo Shibuya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

vi

Dominect: A Simple yet Deep 2-Player Board Game
Oswin Aichholzer, Maarten Löffler, Jayson Lynch, Zuzana Masárová, Joachim Orthaber, Irene Parada,
Rosna Paul, Daniel Perz, Birgit Vogtenhuber and Alexandra Weinberger . . . . . . . . . . . . . . . . . . . 112
Rainbow and Properly Colored Spanning Trees in Edge-Colored
Bipartite Graphs
Mikio Kano and Masao Tsugaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
The Set Chromatic Number of the Middle Graph of Extended Stars
Mark Anthony Tolentino and Gerone Russel Eugenio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
On rainbow mean colorings of brooms and double brooms
Agnes Garciano, Reginaldo Marcelo, Mari-Jo Ruiz and Mark Anthony Tolentino . . . . . . . . . . . . . . 118
Fractional Graph Capacity
Ghurumuruhan Ganesan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Unit Tilt Row Relocation in a SquareDavid Caballero, Angel Cantu, Timothy Gomez, Austin Luchsinger, Robert Schweller and Tim Wylie . . . 122
Discrete Quantum Systems via Tight Group Frames and their Geometrization
Francis Delloro and Job Nable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Direction-Critical Configurations in Noncentral-General Position
Silvia Fernandez and Rimma Hamalainen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Parallel Curves Detection Method Based on Hough Transform
Pat Vatiwutipong and Nattapol Chanpaisit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Computing the Probability of Striking a BattleshipJack Spalding-Jamieson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
The Legend of Zelda: The Complexity of Mechanics
Jeffrey Bosboom, Josh Brunner, Michael Coulombe, Erik D. Demaine, Dylan H. Hendrickson, Jayson
Lynch and Lorenzo Najt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Super Catalan Numbers, Chromogeometry, and Fourier Summations over Finite Fields
Kevin Limanta and Norman Wildberger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Snakes and Ladders with Large Spinners under an Alternative Winning Rule
Waitin Sinthu-Urai and Piyashat Sripratak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Back cover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

vii

Presenters of the talks are indicated by *The 23rd Thailand-Japan Conference on Discrete and Computational Geometry,
Graphs, and Games (TJCDCG3 2020+1)
September 3rd-5th, 2021 (GMT +7)
Department of Mathematics, Faculty of Science, Chiang Mai University
Friday September 3rd, 2021
Time

Activities

09.00 – 09.25

Opening Ceremony• VDO Presentation (Sci CMU – Math CMU)
• Opening Address by
Prof. Torranin Chairuengsri, The Dean of the Faculty of Science, Chiang Mai University
Prof. Jin Akiyama, The Chair of JCDCGGG steering committee• Brief Introduction of TJCDCGGG2020 bySupanut Chaidee, The Co-chair of TJCDCGGG2020
Assoc. Prof. Wanida Hemakul, The Chair of Program Committee

09.25 – 09.30

Group Photo

09.30 – 10.30

Invited Speaker 1: Prof. Kokichi Sugihara
Family Tree of Impossible Objects Created by Optical Illusion
Chair: Supanut Chaidee

10.30 – 10.40

Break

10.40 – 12.00

Parallel sessions 1 (4 talks per room)
Room A
Chair: Teerapong Suksumran

Room B
Chair: Hiro Ito

Room C
Chair: Chao Yang

Xi Shen and Aaron Williams*.
A k-ary middle levels conjecture

Erik Demaine and Kritkorn
Karntikoon*.
Unfolding orthotubes
with a dual Hamiltonian path

Ghurumuruhan Ganesan*.
Redundancy of linear codeswith graph constraints

Sethuraman Guruswamy and
Murugan Varadhan*.
Every tree is a subtreeof a graceful unicyclic graphJosh Brunner, Erik D. Demaine,
Dylan Hendrickson, Victor Luo*and Andy Tockman*.
Complexity of simple folding
orthogonal crease patterns

Nattawut Phetmak*.
Random derangementwith fixed number of cycles

Eduardo Rivera-Campo*.
Graph of uv-paths
in 2-connected graphs

Kota Chida*, Erik Demaine, Martin
Demaine, David Eppstein, Adam
Hesterberg, Takashi Horiyama,
John Iacono, Hiro Ito, Stefan
Langerman, Ryuhei Uehara andYushi Uno.
Multifold tiles of polyominoes
and convex lattice polygons

Tomoaki Abuku* and Masato
Tada.
Multiple hook removing game

Korina Ernjulie Manaloto* andRovin Santos.
Prime labeling of trees
using Eisenstein integers[video presentation]

Joshua Ani, Josh Brunner*, Erik D.
Demaine, Martin L. Demaine,
Dylan Hendrickson, Victor Luo andRachana Madhukara.
Orthogonal fold & cut

12.00 – 13.00

Lunch break

13.00 – 14.00(08.00 HU time:
GMT+2)

Invited Speaker 2: Prof. Stefan Langerman
Towards Dynamic Voronoi Diagrams
Chair: Yushi Uno

14.00 – 14.20

Break

1

Friday September 3rd, 2021
14.20 – 15.40

Parallel sessions 2 (4 talks per room)
Room A
Chair: Kirati Sriamorn

Room B
Chair: Ryuhei Uehara

Room C
Chair: Toshinori Sakai

Johannes Obenaus* and Joachim
Orthaber.
Complete geometric graphswith no partition intoplane spanning trees

Chao Yang*.
Tiling the plane connectively
with Wang tiles

Ilya Bogdanov, Grigory Chelnokov
and Margarita Akhmejanova*.
Moving gold sand game

Zijian Xu and Vorapong
Suppakitpaisarn*.
On the size of minimal separators
for treedepth decomposition

Takashi Yoshino*.
Renzuru tilings
with asymmetric quadrilaterals

Hironori Kiya* and Hirotaka Ono.
Multi-player open-hand
BABANUKI[video presentation]

Chutima Saengchampa* andChariya Uiyyasathian.
Hamiltonian decompositions ofcomplete 4-partite 3-uniform
hypergraphs

Luis Jr. Silvestre* and Job Nable.
Generating frames
via discretized substitution tilings[video presentation]

Koki Suetsugu*.
Playing impartial games
on a simplicial complex
as extension of emperor sum

Rismawati Ramdani, Irsa
Islammeidini Rusvianti and Hasni
Rahmani Rohim.
On the edge irregularity strengthof some disjoint union graphs

Min Yan*, Yohji Akama, Hoi Ping
Luk and Erxiao Wang.
Tiling of the sphereby congruent polygons

Gerard Francis Ortega*.
Losing positions of Splythoffand 2-Splythoff encodedin the Tribonacci word

Saturday September 4th, 2021
Time

Activities

09.00 – 10.00(22.00 NY time:
GMT-4)

Invited Speaker 3: Prof. Erik Demaine
Understanding the Complexity of Motion Planning through Gadgets
Chair: Hiro Ito

10.00 – 10.20

Break

10.20 – 11.20(13.20 AEST time:
GMT+10)Invited Speaker 4: Prof. Daniel Horsley
Decomposing Complete Multigraphs into Stars of Varying Sizes
Chair: Wannasiri Wannasit

11.20 – 12.20

Parallel sessions 3 (3 talks per room)
Room A
Chair: Chariya UiyyasathianRoom B
Chair: Ryuhei Uehara

Room C
Chair: Chao Yang

Md. Manzurul Hasan*, Debajyoti
Mondal and Md. Saidur Rahman.
Linear-time rectilinear drawings oftriconnected subcubic planargraphs with orthogonally convex
faces

Kazuki Matsubara* and Chie Nara.
The maximum numbersof the rigid faces and edges
in continuous flattening processesof a polyhedron

Nóra Frankl, Andrei Kupavskii andArsenii Sagdeev*.
Max-norm analogsof Euclidean Ramsey theorems

Reiya Nosaka*, Hiroyuki Miyata
and Shin-ichi Nakano.
A complete combinatorial
characterizationof greedy-drawable trees

Giovanni Viglietta*, Csaba Tóthand Jorge Urrutia.
Minimizing visible edgesin polyhedra

Lily Chung* and Erik Demaine.
Celeste is PSPACE-hard

Chengyang Qian*.
Number of Go positions
on a connected graph

Hiro Ito and Sae Neshiba*.
Flat folding problem with parallelcreases with mountain-valley
assignment on a convex polygonal
piece of paper

Kazushi Ito and Yasuhiko
Takenaga*.
NP-completenessof peg Solitaire on graphs

12.20 – 13.00

Lunch Break

13.00 – 14.00(08.00 BE time:
GMT+2)

Invited Speaker 5: Prof. Janos Pach

14.00 – 14.20

Break

Crossing Parallels
Chair: Jin Akiyama, Gek Ling Chia

2

Saturday September 4th, 2021
14.20 – 15.20

Parallel sessions 4 (3 talks per room)
Room A
Chair: Ratinan Boonklurb

Room B
Chair: Mari-Jo P. Ruiz

Room C
Chair: Chao Kusollerschariya

Penying Rochanakul* and Sayan
Panma.
Prime-graceful number

Jin Akiyama, Ikuro Sato*.
Distance to antipodeof semi-regular polytope,
measured by edges of equal length

Ryohei Miyadera* and Hikaru
Manabe*.
Previous player's positionsof impartial three-dimensional
chocolate-bar games

Aroonwan Suebsriwichai* andThanasak Mouktonglang.
Rainbow connection numberof Dutch windmill graph

Jin Akiyama, Kiyoko Matsunaga,
Sachiko Nakajima* and Natsumi
Oyamaguchi.
Möbius flowers and buds
[video presentation]

Ryohei Miyadera*, Hikaru
Manabe*, Kousei Suzuki, Taishi
Aono*, Shouei Takahasi and Sohta
Kannan.
Chocolate games andrestricted Nim

Sitta Alief Farihati*, A. N. M
Salman and Pritta Etriana Putri.
Rainbow connection numbersof s-overlapping r-uniform hypertrees

Nattapon Rintaew and Supanut
Chaidee*.
The conditions for fitting a squarein a trapezoid

Kevin Limanta, Hopein Christofen
Tang and Yozef Tjandra*.
Permutation-generated maps
between Dyck paths[video presentation]

Room A
Chair: Gek Ling Chia

Room B
Chair: Chie Nara

Room C
Chair: Yushi Uno

Sujoy Bhore, Jean Cardinal, John
Iacono* and Grigorios Koumoutsos.
Dynamic independent set of squares

Toshinori Sakai*.
Unidirectional monotonic paths
through specified points in labeledpoint sets in convex position[video presentation]

Naoki Matsumoto and Atsuki
Nagao*.
Feedback game on Eulerian graphs

Saharath Sanguanpong* andNantapath Trakultraipruk.
Γ-induced-paired
dominating graphs of cycles

Yiyang Jia*, Jun Mitani and Ryuhei
Uehara.
Logical matrix representations
in map folding

Robert Barish* and Tetsuo
Shibuya.
Solving teleportation mazeswith limited visibility

15.20 – 15.40

Break

15.40 – 16.40

Parallel sessions 5 (3 talks per room)

Tharit Sereekiatdilok* andPanupong Vichitkunakorn.
Biased domination game

Oswin Aichholzer, Maarten
Löffler*, Jayson Lynch, Zuzana
Masárová, Joachim Orthaber, Irene
Parada, Rosna Paul, Daniel Perz,
Birgit Vogtenhuber and Alexandra
Weinberger.
Dominect: a simple yet deep
2-player board game3

Sunday September 5th, 2021
Time
9.00 – 10.20

Activities
Parallel sessions 6 (4 talks per room)
Room A
Chair: Kenta Ozeki

Room B
Chair: Wacharin Wichiramala

Room C
Chair: Hiro Ito

Mikio Kano* and Masao Tsugaki.
Rainbow and properly coloredspanning trees in edge-coloredbipartite graphs[video presentation]David Caballero, Angel Cantu*,
Timothy Gomez, Austin
Luchsinger, Robert Schweller andTim Wylie.
Unit tilt row relocation
in a square[video presentation]

Jack Spalding-Jamieson*.
Computing the probabilityof striking a battleship[video presentation]

Mark Anthony Tolentino* andGerone Russel Eugenio.
The set chromatic numberof the middle graphof extended stars[video presentation]

Francis Delloro* and Job Nable.
Discrete quantum systems
via tight group frames
and their geometrization[video presentation]

Jeffrey Bosboom, Josh Brunner,
Michael Coulombe*, Erik D.
Demaine, Dylan H. Hendrickson,
Jayson Lynch and Lorenzo Najt.
The Legend of Zelda:
The complexity of mechanics

Agnes Garciano, Reginaldo
Marcelo*, Mari-Jo Ruiz and Mark
Anthony Tolentino.
On rainbow mean coloringsof brooms and double brooms

Silvia Fernandez and Rimma
Hamalainen*.
Direction-critical configurations
in noncentral-general position[video presentation]

Kevin Limanta* and Norman
Wildberger.
Super Catalan numbers,
chromogeometry, and Fouriersummations over finite fields

Ghurumuruhan Ganesan*.
Fractional graph capacity

Pat Vatiwutipong and Nattapol
Chanpaisit*.
Parallel curves detection method
based on Hough transform

Waitin Sinthu-Urai* and Piyashat
Sripratak.
Snakes and ladders
with large spinners
under an alternative winning rule10.20 – 10.40

Break

10.40 – 11.40

Invited Speaker 6: Prof. Jittat Fakcharoenphol
Approximation Schemes for Geometric NP-hard Problems: Geometry Meets Algorithms
Chair: Wanida Hemakul

11.40 – 12.00

Closing Ceremony• Thank you speech by
Nattakorn Sukantamala, The Head of the Department of Mathematics• Special issue information by
Supanut Chaidee, The Co-chair of TJCDCGGG2020• Closing speech by
Assoc. Prof. Wanida Hemakul, The Chair of Program Committee
Prof. Jin Akiyama, The Chair of JCDCGGG steering committee4

Abstract(Invited Speakers)

5Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Family Tree of Impossible Objects Created by Optical Illusions
Kokichi Sugihara1
1

MIMS, Meiji University, 4-21-1, Nakano, Nakano-ky, Tokyo 164-8525, Japan
kokichis@meiji.ac.jp (K. Sugihara)
Abstract
This article classiﬁes impossible objects created by Kokichi Sugihara into several categories andlocates them in a form of a family tree according to the reason why they look impossible.
Keywords: optical illusion, impossible ﬁgure, impossible motion, ambiguous object, depth perception.
2010 MSC: Primary 00A66; Secondary 51A05.

1Introduction

Impossible objects were ﬁrst shown by 2D ﬁgures called impossible ﬁgures [2]. When we see thosepictures, we have some ideas of 3D structures, but at the same time we feel they cannot physicallyexist. In this sense, impossible objects were imaginary structures. However, several trickes were foundto construct real 3D structures whose appearances match impossible ﬁgures [1]. In particular, Sugihara
proposed various real 3D objects that look impossible due to optical illusions [3, 4]. In this article, wedeﬁne the term impossible objects as 3D physical objects whose behaviors look impossible, and classify
Sugihara’s impossible objects into a form of a family tree.

2

Impossible Objects and Their Family Tree

There are two main sources of 3D optical illusions. One is the freedom in the choice of the depth froma picture. Illusion occurs when the brain chooses a depth which is diﬀerent from the real depth. The
second is the freedom in the choice of the viewpoint from which we see a 2D image. Illusion occurs when
we see the image from a viewpoint diﬀerent from the center of projection. On the basis of these sourcesof illusions, we can classify the impossible objects.
Figure 1 shows a family tree of impossible objects. Here, the root is at the upper left corner and thebranches extend downward and to the right. The rectangles show the freedom- and perception-related
properties, and rounded boxes represent the classes of impossible objects. A tree branch indicates thata child node is created from the parent node, and the symbols by edges represent how the child nodes
are created. The symbol = means that the same principle is applied to a diﬀerent way of display.
The symbol > means that the principle is strengthened by additional constraints. The symbol +means that two principles are combined to create a new class.
Figure 2 shows examples of objects belonging to each class in the tree.
Acknowledgments. This work is supported by JSPS KAKENHI Grant Nos. 21H03530 and 21K19801.

References[1] G. Ernst, Impossible World, Taschen GmbH, Korn, 2006.
[2] L. S. Penrose and R. Penrose, A special type of visual illusion, British Journal of Psychology, 49, 31–33,
1958.
[3] K. Sugihara, Machine Interpretation of Line Drawings, MIT Press, Cambridge, 1986.
[4] K. Sugihara, Topology-disturbing objects: A new class of 3D optical illusion, Journal of Mathematics andthe Arts, 12, 1–17, 2018.

6

K. Sugihara

Family Tree of Impossible Objects

TJCDCGGG2021<Freedom in Choosing Depth><Preference for Rectangles>

Anomalous Obj. > Ambiguous Obj. (2014~)(1980s~)>=Topology-Disturbing Obj. (2016~)
Impossible Motion Obj.
=
Deformable Obj. (2016~)(1990s~)=
Reflexively-Fused Obj. (2017~)<Freedom in Choosing Viewpoints>>
Ambiguous Tiling (2020~)<Height Reversal Property>>

Height-Reversing Obj.
(2017~)
Triply Ambiguous Obj.
(2018~)+

Partly Invisible Obj. (2015~)
Obj. with Six Interpretations(2019~)+

Figure 1: Family tree of impossible objects.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Figure 2: Examples of impossible objects located in the tree: (a) anomalous object; (b) impossible motion
object; (c) ambiguous object; (d) topology-disturbing object; (e) deformable object; (f) reﬂexibly-fused
object; (g) ambiguous tiling; (h) height-reversing object; (i) partly invisible object; (j) triply ambiguous
obect; (k) object with six interpretations.

7Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Towards Dynamic Voronoi Diagrams
Stefan Langerman∗
Département d’Informatique, Université Libre de Bruxelles, Belgium
stefan.langerman@ulb.ac.be

Abstract
Voronoi Diagrams for a set P of points in the plane associate to each point q ∈ P the regionof the plane that is closer to q than to any other point of P . Together with their dual, Delaunay
Graphs, they are some of the most relevant structures used to understand the distance or proximity
for sets of points. Now what happens when the set P is modified?
It turns out Voronoi Diagrams are remarkably fragile, and the addition, deletion or perturbationof a single point can sometimes change a constant fraction of its features. This makes the goal of
efficiently maintaining Voronoi Diagrams as point sets change or move, seemingly unattainable. But
is it?

∗ Directeur de recherches du F.R.S.-FNRS

8

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Understanding the Complexity of
Motion Planning through Gadgets
Erik D. Demaine1
1

Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
edemaine@mit.edu (E. Demaine)

Abstract
Most motion planning problems — designing the route for one or more agents (robots, humans,
cars, drones, etc.) through a changeable environment — are computationally difficult: NP-hard,
PSPACE-hard, or worse. Such hardness proofs usually consist of several gadgets — local pieces ofenvironment with limited agent interactions/traversals, some of which change local state, which inturn change available interactions/traversals — that can be pasted together into the overall reduction.
In this talk, I’ll describe our quest to characterize exactly which such gadgets suffice to prove different
kinds of hardness, in our motion-planning-through-gadgets framework that has developed over the
past few years [2–6]. This theory enables many hardness proofs, old and new, to be distilled down
to a single diagram of a single gadget.
Keywords: gadgets, motion planning, computational complexity.
2010 MSC: Primary 68Q17; Secondary 68Q25.

Figure 1: An example of a one-figure proof that Pull!-1F(G) (pulling blocks with forced pulling, fixedblocks, and optional gravity) is PSPACE-complete, from JCDCGGG 2019 [1]. This construction implements a self-closing door gadget from the framework of [2].
9

E. D. Demaine

Understanding the Complexity of Motion Planning through Gadgets

TJCDCGGG2021

References[1] Joshua Ani, Sualeh Asif, Erik D. Demaine, Yevhenii Diomidov, Dylan Hendrickson, Jayson Lynch, Sarah
Scheffler, and Adam Suhl, PSPACE-completeness of pulling blocks to reach a goal, Journal of Information
Processing 28 (2020), 929–941.
[2] Joshua Ani, Jeffrey Bosboom, Erik D. Demaine, Yevhenii Diomidov, Dylan Hendrickson, and Jayson Lynch,
Walking through doors is hard, even without staircases: Proving PSPACE-hardness via planar assemblies ofdoor gadgets, Proceedings of the 10th International Conference on Fun with Algorithms (FUN 2020) (La
Maddalena, Italy), September 2020, pp. 3:1–3:23.
[3] Joshua Ani, Erik D. Demaine, Dylan H. Hendrickson, and Jayson Lynch, Trains, games, and complexity:
0/1/2-player motion planning through input/output gadgets, arXiv:2005.03192, May 2020, https://arxiv.
org/abs/2005.03192.
[4] Erik D. Demaine, Isaac Grosof, Jayson Lynch, and Mikhail Rudoy, Computational complexity of motion
planning of a robot through simple gadgets, Proceedings of the 9th International Conference on Fun with
Algorithms (FUN 2018) (La Maddalena, Italy), June 2018, pp. 18:1–18:21.
[5] Erik D. Demaine, Dylan Hendrickson, and Jayson Lynch, Toward a general theory of motion planning complexity: Characterizing which gadgets make games hard, Proceedings of the 11th Conference on Innovationsin Theoretical Computer Science (ITCS 2020) (Seattle, Washington), January 2020, pp. 62:1–62:42.
[6] Jayson Lynch, A framework for proving the computational intractability of motion planning problems, Ph.D.
thesis, Massachusetts Institute of Technology, September 2020.

10

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Decomposing complete multigraphs into stars of varying sizes
Rosalind A. Cameron1 and Daniel Horsley2
1

School of Mathematics and Statistics, University of Canterbury, Christchurch 8140, New Zealand
School of Mathematics, Monash University, Vic 3800, Australia
rahoyte@outlook.com (R.A. Cameron), danhorsley@gmail.com (D. Horsley)2

Abstract
An edge decomposition of a multigraph G is a set of subgraphs of G such that each edge of G occurs
in precisely one of the subgraphs. In 1979 Tarsi showed that an edge decomposition of a complete
multigraph into stars of size m exists whenever the obvious necessary conditions hold. In 1992 Lonc
gave necessary and sufficient conditions for the existence of an edge decomposition of a (simple)
complete graph into stars of sizes m1 , . . . , mt . We are concerned with the common generalisation ofthese problems: when does a complete multigraph admit an edge decomposition into stars of sizes
m1 , . . . , mt ? This problem exhibits more complicated and interesting behaviour than either of its
specialisations. It turns out that the general problem is NP-complete, but that it becomes tractable
if we place a strong enough upper bound on max(m1 , . . . , mt ). We are able to determine the upperbound at which this transition occurs. The basis of this result is a characterisation of when an
arbitrary multigraph can be decomposed into stars of sizes m1 , . . . , mt with specified centres, which
extends a result of Hoffman. A generalisation of Landau’s theorem on tournaments also follows fromthis characterisation.
Keywords: edge decomposition, star, complete multigraph, NP-completeness, tournament.
2010 MSC: Primary 05C51; Secondary 68Q17.

1Background

For a positive integer m, an m-star is a connected simple graph with m edges, all of which are incident
with a single vertex which we call the centre. We denote the λ-fold complete multigraph on n vertices
by λKn . An (edge) decomposition of a multigraph G is a collection of subgraphs of G such that eachedge of G occurs in precisely one subgraph of the decomposition.
In 1979 Tarsi [6] showed that some numerical necessary conditions for the existence of a decompositionof a complete multigraph into stars of a uniform specified size are also sufficient:
Theorem 1.1 (Tarsi [6]). The complete λ-fold multigraph λKn of order n ≥ 3 has a decomposition intom-stars if and only if λ n2 ≡ 0 (mod m) and n
if λ = 1
 2n−1
if λ is even
m≤
 λ
(n
−
1)
if λ ≥ 3 is odd.
λ+1
In 1992 Lonc [5] gave a simple numerical characterisation for the existence of a decomposition of asimple complete graph into stars of various specified sizes:
Theorem 1.2P(Lonc [5]). The
 complete graph Kn has a decomposition into stars of sizes m1 ≥ · · · ≥ mtt
if and only if i=1 mi = n2 and
k
X

mi ≤

n
2



− n−k
2



for each k ∈ {1, . . . , n − 1}.

i=1

11

Cameron and Horsley

2

Decomposing complete multigraphs into stars of varying sizes

TJCDCGGG2021

Our work

Our work [2] addresses the common generalisation of the problems solved by Theorems 1.1 and 1.2:
when does a complete multigraph admit an edge decomposition into stars of sizes m1 , . . . , mt ? Perhaps
surprisingly, this problem exhibits more complicated and interesting behaviour than either of its specialisations. It turns out that the general problem is NP-complete, but that it becomes tractable if we placea strong enough upper bound on max(m1 , . . . , mt ). In fact, we are able to determine the upper boundat which this transition occurs.
To be more precise, we first formalise the question as a family of decision problems, one for eachpositive integer λ and real number α such that 0 ≤ α ≤ 1:(λ, α)-star decomp
Instance: Positive integers n and m1 ≥ · · · ≥ mt such that m1 ≤ α(n − 1) and m1 + · · · + mt = λ n2 .
Question: Does λKn have a decomposition into stars of sizes m1 , . . . , mt ?
Note that m1 + · · · + mt = λ n2 and m1 ≤ n − 1 are obvious necessary conditions for the existence ofthe required decomposition. Then our main result is the following.
Theorem 2.1. Let λ ≥ 2 be an integer. Then (λ, α)-star decomp is NP-complete if α > α0 , where(λ
if λ is odd
λ+1
0
√
α =
1 − λ2 (3 − 2 2) if λ is even.
Furthermore, if α ≤ α0 then every instance of (λ, α)-star decomp is feasible and the required decompositions can be constructed in polynomial time.
The foundation of our proof of Theorem 2.1 is a characterisation of when an arbitrary multigraphcan be decomposed into stars of sizes m1 , . . . , mt where the centre vertex of each star is specified. Thisextends a result of Hoffman [3]. Like Hoffman’s result, our characterisation is proved using the max-flow
min-cut theorem.
We also use our extension of Hoffman’s result to obtain a generalisation of Landau’s theorem on
tournaments [4]. A λ-fold tournament is a λ-fold complete multigraph equipped with an orientationof its edges. A simple extension of Landau’s result (see [1, Theorem 2.2.4], for example) characterises
when there exists a λ-fold tournament of order n with a specified out-degree at each vertex. This
characterisation is in terms of n easily computable inequalities. We achieve an analogous characterisation
when, for each vertex, both the out-degree and a lower bound on the number of distinct out-neighbours
are specified.
Acknowledgments. This work was supported by an AARMS Postdoctoral Fellowship and Australian
Research Council grants DP150100506 and FT160100048.

References[1] R.A. Brualdi, Combinatorial matrix classes, Encyclopedia of Mathematics and its Applications, 108, Cambridge University Press, Cambridge, 2006.
[2] R.A. Cameron and D. Horsley, Decompositions of complete multigraphs into stars of varying sizes. J. Combin.
Theory Ser. B 145 (2020), 32–64.
[3] D.G. Hoffman, The real truth about star designs, Discrete Math. 284 (2004), 177–180.
[4] H.G. Landau, On dominance relations and the structure of animal societies. III. The condition for a scorestructure, Bull. Math. Biophys. 15 (1953), 143–148.
[5] Z. Lonc, Partitions, packings and coverings by families with nonempty intersections, J. Combin. Theory Ser.
A 61 (1992), 263–278.
[6] M. Tarsi, Decomposition of complete multigraphs into stars, Discrete Math. 26 (1979), 273–278.

12Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Crossing Parallels
János Pach
Rényi Institute of Mathematics, Budapest, Hungary
pach@cims.nyu.edu

Abstract
According to the crossing lemma of Ajtai, Chvátal, Newborn, Szemerédi and Leighton (1981),
3
the crossing number of any graph with n vertices and e edges is at least c ne 2 , where c is an absolute
constant. This result, which is tight up to the constant factor, has been successfully applied to avariety of problems in discrete and computational geometry, additive number theory, algebra, andelsewhere. The lemma easily extends to multigraphs: if the multiplicity of every edge is at moste3
m, then the crossing number of the graph is at least c mn
2 . However, much better bounds can beestablished for multigraphs in which no two parallel edges are homotopic. The latest results are joint
with Fox and Suk.

13

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Approximation schemes for geometric NP-hard problems:
geometry meets algorithms
Jittat Fakcharoenphol11Department of Computer Engineering, Kasetsart University, Bangkok 10900, Thailand
jittat@gmail.com (J. Fakcharoenphol)

Abstract
In computer science, numerous optimization problems are NP-hard; thus, it is unlikely to findefficient algorithms for solving them exactly. However, many problems have natural instances on
Euclidean planes, for example the Travelling Salesman Problem, the Minimum Steiner Tree Problem,
covering problems in geometric graphs, and many clustering problems. Geometry gives structures
to these instances, allowing computer scientists to devise efficient approximation schemes that find
(1 + )-approximate solutions in polynomial time for any constant  > 0. In this survey, we outline
many important ideas used in those algorithms, including dynamic programming and randomization.
Keywords: NP-hard problems, geometric instances, approximation schemes.

14Abstract(Contributed Talks)15Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

A k-ary Middle Levels Conjecture
Xi Sisi Shen1 , and Aaron Williams2
1

Department of Mathematics, Columbia University, New York City 10027, USA
Department of Computer Science, Williams College, Williamstown 01267, USA
x.s.shen@columbia.edu (X. S. Shen), aaron.williams@williams.edu (A. Williams)
2

Abstract
We generalize the middle levels conjecture from binary to k-ary strings. Our conjecture is equivalent to star transposition graphs on uniform-frequency multiset permutations being Hamiltonian.
Keywords: Middle levels conjecture, Hamilton cycle, Gray codes, multiset permutations, star transpositions.
2010 MSC: Primary 05C45; Secondary 94C15, 90B99, 90C99.

1

The Middle Levels Conjecture and Theorem(s)A felicitous but unproved conjecture may be of much more consequence for mathematics
than the proof of many a respectable theorem.
— A. Selberg (Collected papers, vol. I (1989), p. 700)

Let Bw (n) be the set of n-bit binary strings with weight w (i.e. with w 1’s) and Bu` (n) = B` (n)∪· · ·∪Bu (n).
The graph G1 (S) has vertex set S and edges between two vertices with Hamming distance one; see Fig. 1a.
A widely renowned combinatorial problem involves this graph on the middle levels strings Bff −1 (2f −1).
Conjecture 1.1 (Middle Levels∗ ). The middle levels graph G1 (Bff −1 (2f −1)) is Hamiltonian when f ≥ 2.
The conjecture was given a difficulty rating of 49/50 by Knuth in The Art of Computer Programming [4], and it was actively worked on for 30 years before a constructive solution by Mütze [7]. The
middle levels conjecture has also provided the starting point for additional research in several directions.
(1) Generalization. The central levels theorem: G1 (Bff +d−1
(2f −1)) is Hamiltonian when d ≥ 1 [2].
−d(2) Strengthening. There are Hamilton cycles in G1 (Bff −1 (2f −1)) with additional symmetries [6].
(3) Variation. De Bruijn sequences for middle and central level strings were constructed [8; 10; 12].
We provide a new generalized conjecture on k-ary strings, and some supporting computational evidence.
00011 00101 00110 01001 01010 01100 10001 10010 10100 11000

100011 100101 100110 101001 101010 101100 110001 110010 110100 111000

00111 01011 01101 01110 10011 10101 10110 11001 11010 11100

000111 001011 001101 001110 010011 010101 010110 011001 011010 011100(a) Middle levels graph G1 (B32 (5)). Note: B32 (5) ' F32 (2, 5).

(b) Star graph G∗ (B3 (6)). Note: B3 (6) ' F3 (2).

Figure 1: (a) Middle levels graph and (b) an isomorphic star graph. They are k = 2 and f = 3 in our conjectures.

2Star Transpositions and Star Graphs

We start by recalling an equivalent formulation of the middle levels problem by Buck and Wiedemann [1].
Given a string s1 . . . sn , a star transposition exchanges s1 and st , thus creating st s2 . . . st−1 s1 st+1 . . . sn .
A star (transposition) graph G∗ (S) has vertices S, and edges joining pairs differing by a star transposition.
Mütze’s middle levels theorem can be restated using star graphs and half-weight binary strings Bf (2f ).
Theorem 2.1 (Star Graph of Half-Weight Binary Strings [7]). G∗ (Bf (2f )) is Hamiltonian when f ≥ 2.
∗ The origin of the middle levels conjecture has been debated, with Havel [3], Dejter, Erdős, and Trotter being mentioned.

16

X. S. Shen et al.

A k-ary Middle Levels Conjecture

TJCDCGGG2021

Proof. The middle levels graph G1 (Bff −1 (2f −1)) is isomorphic to the star graph G∗ (Bf (2f )). In particular, a suitable mapping adds prefix 1 to all v ∈ Bf −1 (2f −1), and 0 to all v ∈ Bf (2f −1); see Fig. 1b.
Let P(k) be the permutations of [k] = {1, 2, . . . , k} in one-line notation. The star graph on permutations is Hamiltonian [5] (even when restrictions are placed on the values that are transposed [11]).
Theorem 2.2 (Star Graph of Permutations [5]). G∗ (P(k)) is Hamiltonian when k ≥ 3.
Now we consider a generalization of Theorems 2.1 and 2.2. Let Ff (k) be the uniform-frequency
6!
= 90
multiset permutations with f copies of each symbol in [k]. For example, {1, 1, 2, 2, 3, 3} has 2!·2!·2!
permutations, so |F2 (3)| = 90. Permutations and half-weight binary strings are special cases: P(k) =F1 (k) and Bf (2f ) ' Ff (2). We also refer to uniform-frequency multiset permutations as uniform strings.
Conjecture 2.3 (Star Graph of Uniform Strings). G∗ (Ff (k)) is Hamiltonian when k ≥ 2 and k + f ≥ 4.
Conjecture 2.3 is false without uniform frequency. For example, G∗ ({112, 121, 211}) isn’t Hamiltonian.
We verified the smallest open case, k = 3 and f = 2, by computer search, with a suitable order in Table 1.
112233 312123 113223 213321 212331 322131 123132 313122 213312 212313 322113 123213 313212 233211 132231
211233 132123 311223 123321 112332 232131 321132 133122 123312 112323 232113 321213 113232 323211 231231
121233 231123 131223 321321 211332 332121 231132 331122 321312 211323 332112 231213 311232 223311 321231
221133 321123 331221 231321 121332 233121 132132 131322 231312 121323 233112 132213 131232 322311 123231
122133 123123 133221 132321 221331 323121 312132 311322 132312 221313 323112 312213 331212 232311 213231
212133 213123 313221 312321 122331 223131 213132 113322 312312 122313 223113 213213 133212 332211 312231

Table 1: Hamilton cycle in the star graph of uniform strings G∗ (F2 (3)) (or 3-ary middle levels graph G1 (F32 (3, 5))).

3

A k-ary Middle Levels Conjecture

Now we restate Conjecture 2.3 using k-ary generalizations of middle level strings and graphs. Let Fu` (k, n)
be the set of frequency range strings of length n, in which each symbol in [k] appears at least ` and at
most u times. In particular, uniform strings are Fff (k, kf ) = Ff (k). By removing one symbol, we obtainthe k-ary middle levels strings Fff −1 (k, kf − 1). These strings have f copies of each symbol in [k], except
any one which will occur f − 1 times. This generalizes the binary middle levels strings, which have f 1’s
and f − 1 0’s, or vice versa. The k-ary middle levels graphs are G1 (Fff −1 (k, kf − 1)); see Figure 1b.
Conjecture 3.1 (k-ary Middle Levels). G1 (Fff −1 (k, kf − 1)) is Hamiltonian when k ≥ 2 and k + f ≥ 4.
Table 1 gives a solution for k = 3 and f = 2, but the first symbol in each string must be omitted.
The original (binary) middle levels problem in Conjecture 1.1 is the k = 2 case of Conjecture 3.1. Adventurous readers may consider Conjecture 3.1 with (1) the central generalization, and/or (2) symmetricstrengthening in Section 1; (3) the k-ary middle levels de Bruijn sequence variation was settled [9; 13].

References[1] M. Buck and D. Wiedemann, Gray codes with restricted density, Discrete Mathematics 48 (1984), 163–171.
[2] P. Gregor, O. Mička, and T. Mütze, On the central levels problem, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, LIPIcs, vol. 168, 2020, pp. 60:1–60:17.
[3] I. Havel, Semipaths in directed cubes, Teubner-Texte zur Mathematik, vol. Bd. 59, 1983, pp. 101–108.
[4] D. E. Knuth, The Art of Computer Programming, vol. 4A: Combinatorial Algorithms, Addison-Wesley, 2011.
[5] V. L. Kompel’makher and V. A. Liskovets, Sequential generation of arrangements by means of a basis oftranspositions, Kibernetica 11 (1975), no. 3, 362–366.
[6] A. I. Merino, O. Micka, and T. Mütze, On a combinatorial generation problem of Knuth, Proceedings of the
ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Jan. 10–13, 2021, SIAM, 2021, pp. 735–743.
[7] T. Mütze, Proof of the middle levels conjecture, Proc. London Math. Soc 112 (2016), no. 4, 677–713.
[8] F. Ruskey, J. Sawada, and A. Williams, De Bruijn sequences for fixed-weight binary strings, SIAM Journal
on Discrete Mathematics 26 (2012), no. 2, 605–617.
[9] J. Sawada and A. Williams, A universal cycle for strings with fixed-content (which are also known as
multiset permutations), WADS 2021: 17th Algorithms and Data Structures Symposium (A. Lubiw and
M. Salavatipour, eds.), Lecture Notes in Computer Science, vol. 12808, Springer-Verlag, 2021.
[10] J. Sawada, A. Williams, and D. Wong, Universal cycles for weight-range binary strings, 24th International
Workshop on Combinatorial Algorithms (IWOCA 2013), Springer, 2013, pp. 388–401.
[11] X. S. Shen and A. Williams, A ‘hot potato’ Gray code for permutations, LAGOS 2013: 7th Latin Conferenceon Graphs and Algorithms, Electronic Notes in Discrete Mathematics, vol. 44, Elsevier, 2013, pp. 89–94.
[12] B. Stevens and A. Williams, The coolest way to generate binary strings, Theory Comput. Syst. 54 (2014),
no. 4, 551–577.
[13] A. Williams, Shift Gray codes, PhD thesis in Computer Science, University of Victoria, 2009.

17

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Every Tree is a Subtree of a Graceful Unicyclic Graph
Sethuraman G1 and Murugan V2

Abstract
Acharya (1982) showed that every connected graph can be embedded in a graceful graph. The
generalization of this result that any set of graphs can be packed into a graceful graph was proved by
Sethuraman and Elumalai (2005). Sethuraman and Ragukumar (2015) have shown that, every tree
can be embedded in a graceful tree. Inspired by these fundamental structural properties of gracefulgraphs, in this paper, we show that every tree can be embedded in a graceful unicyclic graph. Moreprecisely, an algorithm is designed to achieve this embedding.
Keywords: Graceful tree, Graceful unicyclic graph, Graceful tree embedding, Graceful labeling, Graph labeling.
2010 MSC: 05C05, 05C78, 05C85.

1Introduction

In 1963, Ringel posed his celebrated conjecture, popularly called Ringel Conjecture [9], which states that
K2n+1 , the complete graph on 2n + 1 vertices can be decomposed into 2n + 1 isomorphic copies of agiven tree with n edges. In [5], Kotzig independently conjectured the specialized version of the Ringel
Conjecture that the complete graph K2n+1 can be cyclically decomposed into 2n+1 copies of a given treewith n edges. In an attempt to solve both Ringel and Kotzig Conjectures, in 1967, Rosa, in his classicalpaper [10] introduced hierarchical series of labelings called σ, ρ, β and α labelings as a tool to settle both
the Ringel and Kotzig Conjectures. Later, Golomb [4] called β-labeling as graceful labeling, and now
this term is being widely used. A function f is called graceful labeling of a graph G with m edges, if f is
an injective function from V (G) to {0, 1, 2, · · · , m} such that, when every edge uv is assigned the edge
label |f (u) − f (v)|, then the resulting edge labels are distinct. A graph which admits graceful labeling iscalled graceful graph.
Rosa [10] proved the following significant theorem which exhibits the utility of graceful labeling inachieving a cyclic decomposition of the complete graph K2n+1 , where n ≥ 1.
Theorem 1.1. [10] A graph G with n edges admits graceful labeling if and only if the complete graphK2n+1 can be cyclically decomposed into 2n + 1 copies G.
From Theorem 1, one can observe that, if every tree T with n edges admits graceful labeling then thetree T naturally decomposes the complete graph K2n+1 . Theorem 1 led to a famous conjecture, called”Graceful Tree Conjecture”: All trees are graceful. This conjecture remains open over five decades.
Although the structural characterization of graceful graphs appear to be one of the most difficult
problems in graph theory, very few general structural properties have been established. Acharya [1]showed that every connected graph can be embedded in a graceful graph. In [6], Sethuraman andElumalai generalized this result and showed that every set of graphs can be packed into a graceful graph.
Sethuraman and Ragukumar [8] have also shown that every tree can be embedded in a graceful tree.
Recently, [7] Sethuraman and Murugan have shown that every forest can be embedded in a graceful
unicyclic graph. Inspired by these fundamental structural properties of graceful graphs, in this paper,
we show that every tree can be embedded in a graceful unicyclic graph. More precisely, an Algorithm ispresented to achieve this embedding. Our algorithm yields a class of graceful unicyclic graphs generated
from an arbitrary tree. The embedding presented in this paper is totally different and more simpler
than the embedding presented by Sethuraman and Murugan [7]. This result supports the well known
Truszczynski’s Conjecture [11], which claims that, except for the cycles C4m+1 and C4m+2 , all unicycle
graphs are graceful.
18Sethuraman G and Murugan V

2Generating Graceful unicyclic graph

TJCDCGGG2021

Main Result

In this section, we present our Embedding Algorithm, which can embed a given arbitrary tree into agraceful unicyclic graph. More precisely, we construct a graceful unicylic graph from any given tree using
the Embedding Algorithm. Some important procedures of the Embedding Algorithm is given below:
 Labeling the vertices of the given tree T after naming them in a particular order.
 Constructing an intermediate tree T ∗ from the tree T by sequentially adding some labelled pendantvertices.
 Constructing a unicyclic graph G from T ∗ by adding an edge between a chosen pair of non adjacentvertices in T ∗ such that it leads to a graceful labeling of G.

References[1] Acharya B. D, Construction of certain infinite families of graceful graphs from a given graceful graph, Defence
science journal (New Delhi), 32 (1982), 231-236.
[2] David Morgan, Gracefully labeled trees from Skolem sequences, Proc. of The Thirty-First Southeastern Internat. Conf. on Combin., Graph Theory and Computing (Boca Raton, FL, 2000), Congresses Numberantium,
142, 41-48.
[3] Edwards M, Howard L, A survey of graceful trees, Atlantic Electronic J. of Mathematics, 1,1, Summer 2006.
[4] Golomb S. W, How to number a graph, Graph Theory and Computing R.C. Read, ed., Academic Press, New
York, 1972, 23-37.
[5] Kotzig A, Decompositions of a complete graph into 4k-gons (in Russian), Matematicky Casopis, 15 (1965),229-233.
[6] Sethuraman G. and Elumalai A, Packing of any set of graphs into a graceful/harmonious/elegant graph, Ars
Combin.,76 (2005), 297-301.
[7] Sethuraman G. and Murugan V, Generating graceful unicyclic graphs from a given forest, AKCE International Journal of Graphs and Combinatorics 17.1 (2020): 592-605.
[8] Sethuraman G. and Ragukumar P, Towards optimal embedding an arbitrary tree in a graceful tree, Electronic
notes in Discrete Mathematics, 48 (2015), 73-80.
[9] Ringel G, Problem 25, in Theory of Graphs and its Applications, Proc. Symposium Smolenice, Prague, 1963,
page-162.
[10] Rosa A,On certain valuations of the vertices of a graph, Theory of graphs, (International Symposium, Rome,
July 1966), Gordon and Breach, N.Y. and Dunod Paris, 1967, 349-355.
[11] Truszczy´nski M, Graceful unicyclic graphs, Demonstatio Math., 17 (1984), 377–387.
[12] West D. B, Introduction to Graph Theory, Prentice Hall of India, 2nd Edition, 2001.

19

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Graph of uv-paths in 2-connected graphs
Eduardo Rivera-Campo
Departmento de Matemáticas, Universidad Autónoma Metropolitana, Mexico City 09340, Mexico
erc@xanum.uam.mx

Abstract
For a 2-connected graph G we define the uv path graph of G as the graph P(Guv ) whose vertices
are the paths joining u and v in G, where two paths S and T are adjacent if T is obtained from S byreplacing a subpath Sxy of S with an internally disjoint subpath Txy of T . We prove that P(Guv )is always connected and give a necessary condition and a sufficient condition for connectedness inthe case where adjacency between two uv paths S and T is restricted by a set of cycles C where theunique cycle contained in S ∪ T must lie.
Keywords: uv-paths, f -monotone path, Tree Graph.
2010 MSC: Primary 05C38.

For any vertices x, y of a path L, we denote by Lxy the subpath of L that joins x and y. Let G be
a 2-connected graph and u and v be vertices of G. The uv path graph of G is the graph P(Guv ) whose
vertices are the paths joining u and v in G, where two paths S and T are adjacent if T is obtained from S
by replacing a subpath Sxy of S with an internally disjoint subpath Txy of T . The uv path graph P(Guv )is closely related to the graph G(P, f ) of f -monotone paths on a polytope P (see C. A. Athanasiadis etal [1, 2]), whose vertices are the f -monotone paths on P and where two paths S and T are adjacent if
there is a 2-dimensional face F of P such that T is obtained from S by replacing an f -monotone subpathof S contained in F with the complementary f -monotone subpath of T contained in F . We show thatthe graphs P(Guv ) are always connected as is the case for the graphs G(P, f ).
If S and T are adjacent paths in a uv-path graph P(Guv ), then S ∪ T is a subgraph of G consistingof a unique cycle σ joined to u and v by disjoint paths Pu and Pv . Let C be a set of of cycles of G; theuv-path graph of G defined by C is the spanning subgraph PC (Guv ) of P(Guv ) where two paths S and T
are adjacent if and only if the unique cycle σ which is contained in S ∪ T lies in C. A graph PC (Guv )may be disconnected.
The uv path graph P(Guv ) is also related to the well-known tree graph T (G) of a connected graph G,
studied by R. L. Cummins [3], in which the vertices are the spanning trees of G and the edges correspondto pairs of trees S and R which are obtained from each other by a single edge exchange. As in the uv
path graph, if two trees S and R are adjacent in T (G), then S ∪ R is a subgraph of G containing aunique cycle. X. Li et al [5] define, in an analogous way, a subgraph TC (G) of T (G) for a set of cycles C
of G and give a necessary condition and a sufficient condition for TC (G) to be connected. We show thatthe same conditions apply to uv path graphs PC (Guv ).
Similar results are obtained by A. P. Figueroa et al [4] with respect to the perfect matching graph
M(G) of a graph G where the vertices are the perfect matchings of G and in which two matchings L and
N are ajacent if their symmetric difference is a cycle of G. Again, if L and N are adjacent matchings in
M(G), then L ∪ M contains a unique cycle of G.
Keywords: uv-paths, f -monotone path, Tree Graph.
2010 MSC: Primary 05C38.

References[1] C. A. Athanasiadis; J. A. de Loera and Z. Zhang, Enumerative problems for arborescences and monotone
paths on polytopes, arXiv:2002.00999v1 [math.CO].

20

E. Rivera-Campo

Graph of uv-paths

TJCDCGGG2021[2] C. A. Athanasiadis; P. H. Edelman, and V. Reiner, Monotone paths on polytopes, Math. Z. 235 (2000), 315– 334.
[3] R. L. Cummins, Hamilton circuits in tree graphs, IEEE Trans. Circuit Theory, vol. CT-13 (1966), 82–90,[4] A. P. Figueroa; J. Fresán-Figueroa and E. Rivera-Campo, On the perfect matching graph defined by a set ofcycles, Bol. Soc. Mat. Mex. 23 Issue 2 (2017), 549 – 556.
[5] X. Li; V. Neumann-Lara and E Rivera-Campo, On a tree graph defined by a set of cycles, Discrete Math.
271 (2003), 303–310.

21

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Prime Labeling of Trees Using Eisenstein Integers
Korina Ernjulie Manaloto and Rovin Santos
Institute of Mathematics, University of the Philippines, Diliman, Quezon City 1101, Philippines
kemanaloto@math.upd.edu.ph, rbsantos8@up.edu.phAbstract
A tree on n vertices is said to admit a prime labeling if the vertices can be labeled with the firstn natural numbers in such a way that two adjacent vertices have relatively prime labels. The notionof prime labeling has been extended to the Gaussian integers by Lehmann and Park (2016). We
extend this further to the Eisenstein integers by defining an ordering for the Eisenstein integers lying
in the sector 0, π3 of the complex plane. Using this ordering, we give a prime labeling for trees withn ≤ 100 vertices.
Keywords: prime labeling, trees, graph theory, eisenstein integers, eisenstein primes.
2010 MSC: Primary 05C05; Secondary 05C78.

1Introduction

A tree T on n vertices is said to admit a prime labeling if we can label its vertices using the first nnatural numbers in such a way that adjacent vertices have relatively prime labels. In 2002, Pikhurko [4]showed that trees of at most 34 vertices admit a prime labeling under the set of natural numbers without
utilizing computer search algorithms. The notion of prime labeling has been extended to the Gaussian
integers by Lehmann and Park [3] in 2016 using a similar technique. In this paper, we try to achieveanalogous results in prime labeling using the set of Eisenstein integers, that is, complex numbers of thei2πform a + bω, where ω = e 3 and a, b ∈ Z.

2

Preliminaries

2.1

Diagonal Ordering of the Eisenstein Integers

To attain our goal, we begin by introducing the following ordering of the Eisenstein integers lying in the
sector [0, π3 ).
Definition 2.1 (Diagonal Ordering of Eisenstein Integers).
The nth Eisenstein integer γn is defined recursively as follows:
γ1 = 1, and if γn = a + bω for n ∈ N, we have

π
3

8ω
7ω
6ω
5ω
4ω


γn + 1,γ + ω + 1,nγn+1 =
γn + ω,

γn − ω,

if
if
if
if

a ≡ 1 (mod 2) and b = 0
a ≡ 0 (mod 2) and b = a − 1
a ≡ 0 (mod 2) and b 6= a − 1
a ≡ 1 (mod 2) and b 6= 0.

3ω
2ω
ω
0

1

2

3

4

5

6

7

8

Figure 1: Diagonal Ordering of theEisenstein integers.

Figure 1 provides a visual representation of the diagonal ordering of the Eisenstein integers. Somerelevant properties of the diagonal ordering are listed in the following lemmas.
Lemma 2.2. Eisenstein primes are not preceded by their multiples in the diagonal ordering.
Lemma 2.3. Let α be an Eisenstein integer and u be a unit of Z[ω]. Then α and α + u are relatively
prime.
22

K. Manaloto et al.

Prime Labeling

TJCDCGGG2021

As a consequence of Lemma 2.3, we conclude that consecutive Eisenstein integers, as well as odd Eisenstein integers two indices away from each other are relatively prime. Using the preceding properties andthe lemmas presented in the next section, we may now give a prime labeling for trees using Eisenstein
integers which we formally define in the following manner:
Definition 2.4. We say that a tree T = (V, E) on n vertices admits an Eisenstein prime labeling if
we can label its vertices with the first n Eisenstein integers in the diagonal ordering in such a way thatu, v ∈ V have relatively prime labels whenever uv ∈ E.

2.2

Prime Labeling Strategy

The following are the key lemmas in our labeling strategy using the diagonal ordering. The proof of
Lemma 2.5 is analogous to the ordinary integer case, since the only properties needed to establish theproof are Lemma 2.2 and Lemma 2.3. Meanwhile, the proof of Lemma 2.6 can be found in [4].
Lemma 2.5. Let γp be a prime Eisenstein integer and suppose that all trees of order p − 1 admit an
Eisenstein prime labeling under the diagonal ordering. Then all trees of order p, p + 1, and p + 2 also
admit an Eisenstein prime labeling under the diagonal ordering.
Lemma 2.6 (Lemma 2, Pikhurko). Let T be a tree with at least 4 vertices. Then there exists a vertex
u in T such that either
1. there is a subset A of T − u with 3 vertices that can be represented as a union of (one or more)components of T − u; or
2. u has a neighbor v of degree k + 1 ≥ 3 such that its other k neighbors {v1 , v3 , . . . , v2k−1 } each has
degree 2 and are incident to endvertices {v2 , v4 , . . . , v2k } correspondingly.
Lemma 2.5 allows us to exclude from the labeling process trees of order p where γp is prime or those
trees whose order are at most two more than p. For those trees whose orders are not accommodated by
Lemma 2.5, we will use Lemma 2.6 which lets us isolate a branch of the tree that we can easily labelwith large integers in order to reduce the remainder into a smaller tree. The reduction of the tree intoa smaller order will allow us to apply Lemma 2.5 and ensure that it can be labeled using the leftoverEisenstein integers. However, there is no guarantee that the remainder is still a tree. If such situation
occurs, we can use the following result:Lemma 2.7. If any tree on n vertices admit an Eisenstein prime labeling, then any forest on n vertices
also admit an Eisenstein prime labeling.

3

Main Results

Theorem 3.1. All trees of order n ≤ 100 admit an Eisenstein prime labeling under the diagonal ordering.
The proof of the theorem above is done inductively. We start by labeling the tree of order one with γ1 .
We then apply Lemma 2.5 to exclude from the labeling process trees of order p, p + 1 and p + 2, where γp
is prime. The leftover indices are labelled by applying Lemma 2.6. In this strategy, the labeling largely
depends on the distance of the index of the largest prime from the number of vertices of the tree. We
may break these cases into several lemmas, reapplying Lemma 2.6 if necessary. Lastly, we manually label
the cases not covered by the previous process.
Acknowledgments. The authors are grateful to the referees for their careful reading of the manuscript
and their useful comments.

References[1] K.H.A Tout and A.N. Dabboucy, Prime labeling of graphs, National Academy Science Letters, 11 (1982),
pp. 365–368.
[2] J. Gallian, A Dynamic Survey of Graph Labeling, Electron J Combin DS6, 19 (2000).
[3] H. Lehmann and A. Park, Prime Labeling of Small Trees with Gaussian Integers, Rose-Hulman Undergraduate Mathematics Journal, 17 (2016), pp. 73–97.
[4] O. Pikhurko, Every tree with almost 34 vertices is prime, Utilitas Mathematica, 62 (2002).
[5] O. Pikhurko, Trees are almost prime, Discrete Mathematics, 307 (2007), pp. 1455-1462.
[6] D. West, Introduction to Graph Theory, 2nd Edition, Pearson Education, Inc., Delhi, India, 2001.

23

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Unfolding Orthotubes with a Dual Hamiltonian Path
Erik D. Demaine1 and Kritkorn Karntikoon2
1

Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA 02139, USA
Department of Computer Science, Princeton University, Princeton, NJ 08544, USA
edemaine@mit.edu (E. D. Demaine), kritkorn@princeton.edu (K. Karntikoon)
2

Abstract
An orthotube consists of orthogonal boxes (e.g., unit cubes) glued face-to-face to form a path.
In 1998, Biedl et al. showed that every orthotube has a grid unfolding: cutting along edges of theboxes so that the surface unfolds into a connected planar shape without overlap. We give a newalgorithmic grid unfolding of orthotubes with the additional property that the rectangular faces areattached in a single path — a Hamiltonian path on the rectangular faces of the orthotube surface.
Keywords: unfolding polyhedra, orthotubes, Hamiltonicity, algorithm.
2010 MSC: Primary 68Q25; Secondary 52C99.

1

Introduction

Does every orthogonal polyhedron have a grid unfolding, that is, a cutting along edges of the induced grid(extending a plane through every face of the polyhedron) such that the remaining surface unfolds into a
connected planar shape without overlap? This question remains unsolved over 20 years after this type of
unfolding was introduced in 1998 [1]; see [9] for a survey and [3–5] for recent progress. This problem isin some sense the orthogonal nonconvex version of the older and more famous open problem of whether
every convex polyhedron has an edge unfolding (cutting only along edges of the polyhedron) [6].
The first class of orthogonal polyhedra shown to have a grid unfolding is orthotubes [1], formed bygluing together a sequence of orthogonal boxes where every pair of consecutive boxes in the sequenceshare one face (and no other boxes share faces). Roughly speaking, this unfolding consists of a monotone
dual-path of rectangular faces, with O(1) rectangles attached above and below the path.
In this paper, we show that orthotubes have a grid unfolding with a stronger property we call dualHamiltonicity, where the unfolded shape consists of a single dual path of rectangular faces, as shown in
Figure 1. More precisely, define the face adjacency graph to have a node for each rectangular face ofa box, and connect two nodes by an edge whenever the corresponding rectangular faces share an edge.
Then the unfolding is given by keeping attached the duals of edges that form a Hamiltonian path in theface adjacency graph. Implicitly, we take advantage of the fact that 4-connected planar graphs (which
includes face adjacency graphs) have Hamiltonian cycles [2, 10].















(a) orthotube



















 
 
   (b) face adjacency graph(c) dual-Hamiltonian
unfolding
















(d) dual Hamiltonian path

Figure 1: An example of an orthotube, its face adjacency graph, dual-Hamiltonian unfolding corresponding to chain code RSSLRLRL, and the corresponding dual Hamiltonian path.

24

E. D. Demaine et al.

2Unfolding Orthotubes with a Dual Hamiltonian PathTJCDCGGG2021

Unfolding

Our main result is the following:
Theorem 2.1. Given an orthotube, there is a Hamiltonian path of its face adjacency graph such that,
if we follow the path, we get a non-overlapping unfolding.
To prove the result, we use a “chain code” (similar to [7, 8]) to represent a path on the surface of an
orthotube. A chain code is an ordered sequence of the form a1 a2 . . . an , where ai ∈ {L, R, S} represents
an (intrinsic) left turn, right turn, or continuing straight to move from the ith face to the (i + 1)st face.
PnFor each chain code c = a1 a2 . . . an , we define the cumulative quarter turning qturn(c) to be
i=1 qturn(ai ) where qturn(R), qturn(L), and qturn(S) are +1, −1, and 0 respectively. To guarantee that our unfolding does not overlap, we prove the invariant that the qturn of any prefix of the chain
code is in {−1, 0, 1}, that is, the unfolding proceeds monotonically in one direction.

2.1

Algorithm

At a high level, the algorithm creates a chain code for unfolding an orthotube up to three boxes at atime, to maintain the stronger condition that the unfolding so far has qturn = 0. We divide into cases
based on each box’s relative position to the next one, two, and sometimes three or four boxes (if they
exist). Figure 2 shows two cases.

(a) RLRL(b) RSLR

Figure 2: Two possible chain codes, RLRL and RSLR, to unfold the current box from the black face,when the next box (white) is on top. The first code preserves qturn, so we use it unless the next nextbox (not drawn) is attached on the face that we try to enter on the next box (white). The second code
adds 1 to qturn, and we show how in all cases to add an additional box to restore qturn to 0.

References[1] Therese Biedl, Erik Demaine, Martin Demaine, Anna Lubiw, Mark Overmars, Joseph O’Rourke, Steve
Robbins, and Sue Whitesides, Unfolding some classes of orthogonal polyhedra, Proceedings of the 10th
Canadian Conference on Computational Geometry, 1998, pp. 70–71.
[2] Norishige Chiba and Takao Nishizeki, The Hamiltonian cycle problem is linear-time solvable for 4-connectedplanar graphs, Journal of Algorithms 10 (1989), no. 2, 187–211.
[3] Mirela Damian, Erik Demaine, Robin Flatland, and Joseph O’Rourke, Unfolding genus-2 orthogonal polyhedra with linear refinement, Graphs and Combinatorics 33 (2017), no. 5, 1357–1379.
[4] Mirela Damian and Robin Flatland, Unfolding polycube trees with constant refinement, Computational Geometry: Theory and Applications 98 (2021), 101793.
[5] Mirela Damian and Robin Y. Flatland, Unfolding low-degree orthotrees with constant refinement, Proceedingsof the 30th Canadian Conference on Computational Geometry (Winnipeg, Canada), 2018, pp. 189–208.
[6] Erik D. Demaine and Joseph O’Rourke, Geometric folding algorithms: Linkages, origami, polyhedra, reprinted., Cambridge University Press, New York, NY, USA, 2008.
[7] Eduardo Lemus, Ernesto Bribiesca, and Edgar Garduño, Representation of enclosing surfaces from simple
voxelized objects by means of a chain code, Pattern Recognition 47 (2014), no. 4, 1721–1730.
[8], Surface trees – representation of boundary surfaces using a tree descriptor, Journal of Visual Communication and Image Representation 31 (2015), 101–111.
[9] Joseph O’Rourke, Unfolding orthogonal polyhedra, Surveys on Discrete and Computational Geometry:
Twenty Years Later (J. E. Goodman, J. Pach, and R. Pollack, eds.), American Mathematical Society,2008, pp. 231–255.
[10] Carsten Thomassen, A theorem on paths in planar graphs, Journal of Graph Theory 7 (1983), no. 2, 169–176.

25

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Complexity of Simple Folding Orthogonal Crease Patterns
Josh Brunner1 , Erik D. Demaine1 , Dylan Hendrickson1 , Victor Luo1 , and Andy Tockman1
1

MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA 02139, USA
brunnerj@mit.edu (J. Brunner), edemaine@mit.edu (E. Demaine), dylanhen@mit.edu (D. Hendrickson),vluo@mit.edu (V. Luo), tockman@mit.edu (A. Tockman)

Abstract
Continuing results from JCDCGGG 2016 and 2017, we solve several new cases of the simple
folding problem — deciding which crease patterns can be folded flat by a sequence of (some modelof) simple folds. We give new efficient algorithms for mixed crease patterns, where some creases
are assigned mountain/valley while others are unassigned, for all 1D cases and for 2D rectangularpaper with one-layer simple folds. By contrast, we show strong NP-completeness for mixed crease
patterns on 2D rectangular paper with some-layers simple folds, complementing a previous result
for all-layers simple folds. We also prove strong NP-completeness for finite simple folds (no matterthe number of layers) of unassigned orthogonal crease patterns on arbitrary paper, complementing
a previous result for assigned crease patterns, and contrasting with a previous positive result for
infinite all-layers simple folds. In total, we obtain a characterization of polynomial vs. NP-hard for
all cases — finite/infinite one/some/all-layers simple folds of assigned/unassigned/mixed orthogonalcrease patterns on 1D/rectangular/arbitrary paper — except the unsolved case of infinite all-layers
simple folds of assigned orthogonal crease patterns on arbitrary paper.
Keywords: computational origami, simple folding, NP-hardness, algorithms.
2010 MSC: Primary 68Q25; Secondary 52C99.

1

Introduction

In the well-studied simple foldability problem [1–3], we are given a crease pattern consisting of linesegment creases, possibly assigned mountain or valley, on a 2D region called the piece of paper, and areasked whether all of the creases can be folded via a sequence of “simple folds”. Each simple fold folds
some set of layers of the piece of paper around a single line by ±180◦ (thus preserving flatness of thefolding). In the 1D variation, the piece of paper is a 1D line segment and the creases are points.
Many different models for simple folds and special cases for the simple foldability problem have been
considered, depending on the following characteristics:• How many layers of paper a simple fold can move at once. The most powerful model, some-layers,
allows the top or bottom k layers to be folded for any k. Two more restrictive models are one-layer,
which permits folding only a single layer at a time (modeling thick material); and all-layers, which
requires folding all layers simultaneously.
• Whether simple folds can be along finite line segments (chords of the crease pattern), or must be
along infinite lines (modeling a half-plane/large flipping tool).
• Whether some creases are assigned as needing to be folded mountain or valley. In the more common
assigned and unassigned cases, all creases either have or lack an assignment, while in the mixed
case [1], each crease may be mountain, valley, or unassigned.
• What shape the paper is allowed to have. In 1D, the only option is a line segment, while in 2D thetwo cases considered are rectangles and arbitrary polygons, though all NP-hardness results holdeven for simple orthogonal polygons.
26

J. Brunner et al.

Complexity of Simple Folding Orthogonal Crease Patterns

Arbitrary Paper
Infinite&
Finite[2]
NP-comp.
NP-comp. [2]⇓⇓NP-comp.
NP-comp.
⇑⇑
NP-comp. [2]NP-comp. (R4)⊂

Rectangular

One
Layer

poly
⇐
⇑
Mixed
poly (R1) ⇐⇓Unassigned
poly⇐

poly [3]⇑poly (R1)⇓poly [3]Some
Layerspoly⇐
poly [3]NP-comp. [2]NP-comp. [2]
⇑
⇓
⇓
Mixed
poly (R1)
NP-comp. (R3) ⇒ NP-comp. &NP-comp.
⇓⇑⇑
Unassigned
poly⇐
poly [3]NP-comp. [2]NP-comp. (R4)All
Layers

poly
⇐
⇑
Mixed
poly (R2)⇓
Unassigned
poly
⇐

1D
Assigned⊂⊂⊂

TJCDCGGG2021

Assigned⊂⊂

Assignedpoly [3]

OPEN

⊂

⇒

NP-comp.

poly [3]⇐poly [1]⊂NP-comp. [1]NP-comp. [2]⇓&NP-comp.
⇑
NP-comp. (R4)

Table 1: Summary of past results (cited) and new results (bold, with result numbers) about simple folding
orthogonal crease patterns. “NP-comp.” denotes strong NP-completeness. “⊂” denotes containment
between classes of instances. “⇒” denotes implications between results. “R” numbers refer to the results
list in Section 2. Hardness results for arbitrary paper hold even for paper a simple orthogonal polygon.
We consider exclusively orthogonal crease patterns, which contain only vertical and horizontal creases.
Prior work [2, 3] has also studied versions of the problem including diagonal creases.
Table 1 presents all known results according to these parameters, including both previously known
results and new results from this paper (in bold). Note that, for orthogonal crease patterns, the finitevs. infinite simple fold distinction only plays a role with arbitrary paper: no distinction can be made in
the 1D case, and equivalence for rectangular paper is given by [2, Theorem 8].

2

Results

Our new results can be summarized as follows:
1. One- & some-layers mixed 1D; one-layer mixed rectangular. We adapt arguments from [3],and provide a new characterization of when a 1D mixed crease pattern is flat-foldable. For one-layer
rectangular, crossing creases can never be folded, so the problem reduces to 1D paper.
2. All-layers mixed 1D. We show that folding the superficially foldable (foldable ignoring assignments) crease nearest an end of the paper never makes the crease pattern unfoldable, giving anefficient greedy strategy for deciding simple foldability.
3. Some-layers mixed rectangular. We show that the NP-hardness reduction from [1] for all-layers
mixed rectangular also works for some-layers.
4. Unassigned arbitrary finite. We modify the NP-hardness proof from [2] for assigned crease
patterns to work for unassigned crease patterns, by adding a component that enforces the relevant
aspects of the assignment. The original proof and our adaptation apply to any number of layers.
Notably, the only case that remains unsolved for orthogonal crease patterns is infinite all-layers simple
folds of assigned crease patterns on arbitrary paper.

References[1] Hugo A. Akitaya, Cordelia Avery, Joseph Bergeron, Erik D. Demaine, Justin Kopinsky, and Jason S. Ku,
Infinite all-layers simple foldability, Graphs and Combinatorics 36 (2020), 231–244, Special issue from JCDCGGG 2017.
[2] Hugo A. Akitaya, Erik D. Demaine, and Jason S. Ku, Simple folding is really hard, Journal of Information
Processing 25 (2017), 580–589, Special issue from JCDCGGG 2016.
[3] Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Martin L. Demaine, Joseph S. B. Mitchell, Saurabh
Sethia, and Steven S. Skiena, When can you fold a map?, Computational Geometry 29 (2004), no. 1, 23–46.

27

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Multifold tiles of polyominoes and convex lattice polygons
Kota Chida1 , Erik Demaine2 , Martin Demaine2 , David Eppstein3 ,Adam Hesterberg2 , Takashi Horiyama4 , John Iacono5 , Hiro Ito1 ,Stefan Langerman5 , Ryuhei Uehara6 , and Yushi Uno7
1

The University of Electro-Communications, 2 Massachusetts Institute of Technology, 3 University of California,
Irvine, 4 Hokkaido University, 5 Université libre de Bruxelles, 6 Japan Advanced Institute of Science andTechnology, 7 Osaka Prefecture University
k.chida@mail.uec.jp (K. Chida), edemaine@mit.edu (E. Demaine), mdemaine@mit.edu (M. Demaine),
eppstein@ics.uci.edu (D. Eppstein), achester@mit.edu (A. Hesterberg), horiyama@ist.hokudai.ac.jp(T. Horiyama), john@johniacono.com (J. Iacono), itohiro@uec.ac.jp (H. Ito), Stefan.Langerman@ulb.ac.be(S. Langerman), uehara@jaist.ac.jp (R. Uehara), uno@cs.osakafu-u.ac.jp (Y. Uno)

Abstract
A family of 2-dimensional shapes T is called a tiling if they (rotating and reﬂecting are allowed)
cover the whole plane without gaps or overlaps, and if all shapes belonging to T are congruent eachother, then the shape is called a tile. We study k-fold tilings which are extended to cover the planewith the thickness of k folds and k-fold tiles which belong to it. Intuitively it means that a familyof 2-dimensional shapes covers the plane such that they overlap k times at any point in the plane.
Since clearly a (1-fold) tile is a k-fold tile for any positive integer k, the subjects of our researchare 2-dimensional shapes with property “not a tile, but a k(≥ 2)-fold tile.” We call a plane shapesatisfying this property a nontrivial k-fold tile. In this talk, we clarify some facts as follows: ﬁrst,
we show that for any integer k ≥ 2, there exists a polyomino satisfying a property that “not a h-foldtile for any positive integer h < k, but a k-fold tile.” We also ﬁnd for any integer k ≥ 2, polyominoes
with the minimum number of cells among ones which are nontrivial k-fold tiles. Next, we prove thatfor any integer k = 5 or k ≥ 7, there exists a convex unit-lattice polygon with an area of k that is a
nontrivial k-fold tile.
Keywords: multiple tilings, k-fold tiles, polyominoes, convex polygons, thickness.
2010 MSC: Primary 52C20; Secondary 05B45, 05B50.

1Introduction

A family of 2-dimensional shapes T is called a tiling if they (rotating and reﬂecting are allowed) cover
the whole plane without gaps or overlaps. It is said that a tiling T is monohedral [3] if any two shapes
belonging to T are congruent. If a tiling T is monohedral, the unique shape in the tiling is called a tile.
In this article, when we simply say a “tiling,” it means a monohedral tiling. We study (monohedral)k-fold tilings which are extended to cover the plane with the thickness of k folds and k-fold tiles which
belong to it. More speciﬁcally, a family of 2-dimensional shapes T is called a k-fold tiling if any point
in R2 which is not on the boundary of any shape in T is covered with exactly k shapes in T , and theshape belonging to T is called a k-fold tile. Since we can obtain a k-fold tiling by piling up k sheets of atiling, it is trivial to consider a k-fold tiling with (1-fold) tiles. Hence we are interested in 2-dimensional
shapes with the property: not a tile, but a k(≥ 2)-fold tile. We call a 2-dimensional shape having thisproperty a nontrivial k-fold tile.
If all shapes in k-fold tiling T = {T1 , T2 , . . .} are translates of T1 , then T is called a k-fold translativetiling [5, 6], and the shape belonging to T is called a k-fold translative tile [5, 6]. The origin of thestudy of multiple tilings is the one by Furtwängler [2] in 1936. He considered trivial multiple tilings asa generalization of Minkowski’s conjecture [4]. As far as we know, nontrivial multiple tilings were ﬁrst
investigated by Bolle [1] who gave a necessary and suﬃcient conditions for a convex polygon to constructa multiple translative tiling with regularity (called a multiple lattice tiling [5, 6]). Recently, Yang andZong [5] gave a characterization of all convex k-fold translative tiles for any k = 2, 3, 4, 5. Although
28

K. Chida et al.

Multifold tiles of polyominoes and convex lattice polygons

TJCDCGGG2021

there is various research on multiple translative tilings other than those mentioned above, there seems
to be no existing research on multiple tilings allowing rotations and reﬂections. Therefore, the subjectsof our research is such nontrivial multiple tilings. In this article, we mainly consider polyominoes: plane
shapes formed by joining one or more congruent squares edge to edge, and convex lattice polygons asbasic 2-dimensional shapes and present some properties.

2

Preliminaries

We introduce the following terms.
Deﬁnition 2.1. If a shape P is a k-fold tile, then k is a tile-fold number of P . The set of tile-foldnumbers of P is denoted by TFN(P ). If an integer k satisﬁes that k ∈ TFN(P ) and h ∈/ TFN(P ) for
every positive integer h < k, then we call k the minimum tile-fold number of P , and it is denoted byτ • (P ) [6].
Deﬁnition 2.2. Let k and h be positive integers. If an h-omino P is a nontrivial k-fold tile and there isno h′ -omino that is a nontrivial k-fold tile for any positive integer h′ < h, then h is called the minimum
size of nontrivial k-fold-tile polyomino and P is called a minimum-sized nontrivial k-fold-tile polyomino.

3

Main Results

We show the following theorems.
Theorem 3.1. For any integer k ≥ 2, there exists a polyomino P that satisﬁes τ • (P ) = k.
Theorem 3.2. For any integer k ≥ 2, the minimum size of nontrivial k-fold-tile polyomino is 7, and theheptominoes C7, F7, and X7 listed in Fig. 1 are all minimum-sized nontrivial k-fold-tile polyominoes.
Furthermore, the heptomino G7 in Fig. 1 is also a minimum-sized nontrivial k-fold-tile polyomino forevery k ≥ 2 except for k = 3, 5.

Figure 1: The heptominoes in Theorem 3.2
Theorem 3.3. For any integer k = 5 or k ≥ 7, there exists a convex unit-lattice polygon that satisﬁes(i) a nontrivial k-fold tile and(ii) the area is k.
(iii) it is a hexagon if k = 5 or 8; an octagon otherwise.
Acknowledgments. We would like to express our gratitude to Professor Andrew Winslow in University of Texas Rio Grande Valley and Assistant Professor Rémy Belmonte, Mr. Kyohei Chiba, andMr. Kazuki Higashizono in The University of Electro-Communications for their precious discussions. We
also acknowledge the participants of the 32nd Bellairs Winter Workshop on Computational Geometryon January 27–February 3, 2017 in Holetown, Barbados where we had a preliminary discussion on these
problems.

References[1] U. Bolle, On multiple tiles in E 2 , In: Intuitive geometry (Szeged, 1991), Colloq. Math. Soc. János Bolyai,
vol. 63, pp. 39–43, North-Holland, Amsterdam, 1994.
[2] P. Furtwängler, Über Gitter konstanter Dichte, Monatsh. Math. Phys. 43 (1936), 281–288.
[3] J.E. Goodman, J. O’Rourke, C.D. Tóth (eds.), Handbook of discrete and computational geometry, CRC
Press, Boca Raton, 2017.
[4] H. Minkowski, Diophantische approximationen, B. G. Teubner, Leipzig, 1907.
[5] Q. Yang and C. Zong, Characterization of the two-dimensional ﬁvefold translative tiles, Bull. Soc. Math.
France 149 (2021), 119–153.
[6] C. Zong, Can you pave the plane with identical tiles?, Notices Amer. Math. Soc. 67 (2020), 635–646.

29

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Orthogonal Fold & Cut
Joshua Ani1 , Josh Brunner1 , Erik D. Demaine1 , Martin L. Demaine1 ,Dylan Hendrickson1 , Victor Luo1 , and Rachana Madhukara1
1

MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA 02139, USA
joshuaa@mit.edu (J. Ani), brunnerj@mit.edu (J. Brunner), edemaine@mit.edu (E. Demaine),mdemaine@mit.edu (M. Demaine), dylanhen@mit.edu (D. Hendrickson), vluo@mit.edu (V. Luo),rachanam@mit.edu (R. Madhukara)
Abstract
We characterize the shapes that can be produced by “Orthogonal Fold & Cut”: folding a rectangular sheet of paper on vertical and horizontal creases, and then making a single straight cut. We
also solve a handful of simpler related problems: Orthogonal Fold & Punch, 1-Dimensional Fold &Cut, Signed 1-Dimensional Fold & Cut, and 1-Dimensional Interval Fold & Cut.
Keywords: computational origami, folding, cutting, orthogonal, characterization.
2010 MSC: Primary 68Q25; Secondary 52C99.

1

Introduction

Given a rectangular piece of paper marked with some line segments called cuts, Fold & Cut asks us
to fold the paper into a flat configuration and then make a single infinite cut to cut exactly along themarked line segments. More formally, we wish to fold the paper such that the cuts all lie on the sameline, and no other part of the paper is on the line. It turns out that Fold & Cut is solvable for any (finite)
set of line segments [2, 3].
Orthogonal Fold & Cut asks the same question, but weare now restricted to orthogonal folds, meaning every crease
must be parallel to an edge of the rectangle. Again we use
only a single cut, which may be at any angle. It is easy toconstruct sets of cuts which cannot be obtained this way,
but some complicated-looking shapes still can be. Figures 1
Figure 1: An example instance of Orthogand 2 show example instances and their solutions.
onal Fold & Cut (bold green lines), andWe characterize when Orthogonal Fold & Cut is solvthe crease pattern our algorithm generates
able. In addition, we provide characterizations for several(thin red mountains and blue valleys).
simpler related problems:• Given a line segment of paper with marked cut points, 1-Dimensional Fold & Cut asks us to foldthe line segment and make a single (0-dimensional) cut that hits exactly the cut points. We do notallow folds at cut points.
• In Signed 1-Dimensional Fold & Cut, each cut point is given a sign, either positive or negative,
and we are asked to have the paper oriented according to the sign at each cut point. Intuitively,
the cut points are marked on only one side, and they need to all be face up.

Figure 2: A few letters from our Orthogonal Fold & Cut mathematical font, in the notation of Figure 1.
30

J. Ani et al.

Orthogonal Fold & Cut

TJCDCGGG2021• In 1-Dimensional Interval Fold & Cut, we are asked to fold a line segment so that specified cut
intervals lie on a common interval in the folded state, and no other part of the line segment lieson this interval. Additionally, the cut intervals contain some marked creases which are required to
be folded, and no other folds can be made within any cut interval.
• In Orthogonal Fold & Punch, we are given a rectangular piece of paper marked with points called
holes, and are asked to fold it flat using only orthogonal folds and punch out a single point toremove exactly the holes. The nonorthogonal Fold & Punch problem always has a solution [1].

2Results

Our characterizations take the form of defining an easy-to-compute “canonical” crease pattern andshowing that, if the problem has any solution, then the canonical one works.
Lemma 2.1. 1-Dimensional Fold & Cut is always solvable.
Lemma 2.2. Signed 1-Dimensional Fold & Cut is solvable if and only if the signs of cuts alternate.
Refer to Figure 3. The canonical crease pattern for Signed and unsigned 1D Fold & Cut puts a crease
point at the midpoint between each consecutive pair of cut points. For 1D Interval Fold & Cut, thecanonical crease pattern has a crease at each required crease and at the midpoint between each pair ofconsecutive cut intervals.
Lemma 2.3. An instance of 1-Dimensional Interval Fold & Cut is solvable if and only if the canonicalcrease pattern is a solution.

Figure 3: The canonical solution for an instance of 1-Dimensional Fold & Cut, and the canonical solution
for an instance of 1-Dimensional Interval Fold & Cut.

Figure 4: An instance of Orthogonal Fold & Cut with vertical bands shaded (left), and its (unsigned)
canonical crease pattern (right).
For the main problem, Orthogonal Fold & Cut, we first note that the slopes of cut lines in a solvableinstance must all be ±α for a common α. The degenerate cases α = 0 and ∞ are easy, and otherwise we
can scale the paper so α = 1, as in Figure 4. Next, local constraints imply that some intervals, which wecall bands, must not have vertical creases, and that there are vertical creases at some particular positions.
We call other intervals stripes, including zero-width stripes at positions required to have creases. Our
main result characterizes the solvable instances of Orthogonal Fold & Cut in terms of stripes and bands,
by factoring the instance into two othogonal instances of 1D Interval Fold & Cut.
Theorem 2.4. Consider an instance of Orthogonal Fold & Cut on rectangular paper in which every
crease has finite nonzero slope ±α. We call the crease pattern which puts one vertical (resp. horizontal)
crease at the center of each vertical (resp. horizontal) stripe, including zero-width stripes, the canonicalcrease pattern. If the instance is solvable, then the canonical crease pattern is a solution.

References[1] Yasuhiko Asao, Erik D. Demaine, Martin L. Demaine, Hideaki Hosaka, Akitoshi Kawamura, Tomohiro Tachi,
and Kazune Takahashi, Folding and punching paper, Journal of Information Processing 25 (2017), 590–600.
[2] Marshall Bern, Erik Demaine, David Eppstein, and Barry Hayes, A disk-packing algorithm for an origami
magic trick, Origami3 : Proceedings of the 3rd International Meeting of Origami Science, Math, and Education,
A K Peters, 2001, pp. 17–28.
[3] Erik D. Demaine, Martin L. Demaine, and Anna Lubiw, Folding and cutting paper, Revised Papers from theJapan Conference on Discrete and Computational Geometry (JCDCG’98), vol. 1763, pp. 104–117.

31

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Redundancy of Linear Codes with Graph Constraints
Ghurumuruhan Ganesan11Institute of Mathematical Sciences, HBNI, Chennai, India
gganesan82@gmail.com

Abstract
In this paper we use the probabilistic method to obtain bounds for the redundancy of linear codeswith graphical constraints and determine sufficient conditions for existence of such codes.
Keywords: Linear Codes, Graphical Constraints, Redundancy, Probabilistic Method.
2010 MSC: Primary 05C90; Secondary 05C50.

1Introduction

Linear codes with graph based constraints arise often in applications like for example, low density parity
check (LDPC) codes and expander codes [1,3] that requires that the bipartite graph representation satisfy
an expansion property with respect to the left nodes. In this paper, we consider generalized constraints
and use random bipartite graphs to obtain linear codes that attain the Gilbert-Varshamov redundancy
bound in addition to satisfying the said constraints.

2

Linear Codes With Graph Constraints

Consider a random bipartite graph with left vertex set X = {a1 , a2 , . . . , an } and right vertex set Y ={b1 , b2 , . . . , bm } obtained as follows. Let {Xi,j }1≤i≤n,1≤j≤m be independent and identically distributed(i.i.d.) binary random variables with P(X1,1 = 1) = p = 1 − P(X1,1 = 0) where 0 < p < 21 is a constant.
Throughout, constants do not depend on n. An edge is present between vertices ai and bj if and only
if Xi,j = 1. Let m = n for some constant  > 0 and let G = Gn be the resulting random graph definedon the probability space (Ωn , Fn , Pn ). For simplicity we drop the subscript from Pn henceforth.
For 1 ≤ j ≤ m let Rj be the set of neighbours of the rightPvertex bj and define the (random) code C =C(G) as follows. A word c = (c1 , . . . , cn ) ∈ C if and only if i∈Rj ci = 0 mod 2 for all 1 ≤ j ≤ m. By
construction (see [1]), the code C is linear (i.e. a vector subspace of {0, 1}n ) with redundancy≤m
1− log(#C)
nn = . We define the distance between c = (c1 , . . . , cn ) and d = (d1 , . . . , dn ) to be d(c, d) :=
P
n
1(c
1
=
6
d
1 refers to the indicator function. We also define the minimum distance dH (C)ii ), where 1(.)
i=1of the code C to be the minimum distance between any two codewords of C and set the relative distanceof C to be δH (C) := dH (C)−1.
n
An n−length constraint
En on the random code C is an event in Fn . For example, the event Hn√that for each 2 ≤ i ≤ n the right vertices bi−1 and bi+1 both have the left vertex ai as a neighbour
is an example of a constraint. We say that the random code C satisfies the constraint En if G ∈ En .
Letting H(x) := −x · log x − (1 − x) · log(1 − x) be the (binary) entropy function, where logarithms areto the base two throughout and using the notation o(1) −→ 0 as n → ∞, we have the following result.
n)
Theorem 2.1. Let 0 < δ < 21 be any constant and let En , n ≥ 1 be constraints satisfying log P(E= o(1).
n
For all n large, there exists a deterministic n−length linear code Dn with relative distance at least δ,
redundancy at most H(δ) + o(1) and satisfying the constraint En .

Thus if the constraints are not too severe, then there are codes that achieve the Gilbert-Varshamov
redundancy bound [2] of H(δ)+o(1) in addition
to satisfying the said constraints. The event Hn described√
before occurs with probability at least p2 n and therefore satisfies the conditions of Theorem 2.1.
32

G.Ganesan.

TJCDCGGG2021

Proof of Theorem 2.1: The proof of Theorem 2.1 consists of two steps. In the first step, we show thatthe minimum distance of the random code C obtained above is at least δn + 1 with high probability i.e.,with probability converging to one as n → ∞. We then incorporate the constraints into C to obtain thedesired code. Throughout we let  > H(δ) be a constant and let m = n be the number of parity (right)
nodes in the graph G so that the size of C is at least 2n(1−) .
1
if i ∈ S
For a set S ⊆ {1, 2, . . . , n} we define the word v(S) = (v1 , . . . , vn ) satisfying vi =0 otherwise.
Letting S = {al1 , . . . , alg } be any set of left vertices we upper bound the probability that S would cause
no parity check violations; i.e. we estimate the probability that the word v(S) belongs to the code C.
We consider two cases depending on whether #S ≤ t or not, for some integer constant t ≥ 1 to be
determined later.
Case I (#S = g ≤ t): For 1 ≤ i ≤ g let Ni be the set of neighbours of the left vertex ali . Let Mi :=S
Ni \be the set of (unique) neighbours of vertex ali not adjacent to any of the remaining
1≤l≤t Nll6=ivertices in S\{ali }. Defining Yi,j := 11(bj ∈ Mi ) we then have that {Yi,j }1≤j≤m are i.i.d. for any 1 ≤ i ≤ g,
m
g−1
g−1
with P(Yi,j = 1) = p(1= 1 − P(YP (Mi = ∅) = 1 − p(1 − p)g−1≤ e−mp(1−p)
S− p)
 i,j = 0). Thusg−1T−mp(1−p)and consequently P. If the event E (S) := 1≤i≤g {Mi 6= ∅} occurs,
1≤i≤g {Mi = ∅} ≤ geTthen the word v(S) ∈/ C. Therefore if the event Elow := S E (S) occurs where the intersection is withrespect to all subsets S ⊂ {1, 2, . . . , n} of size g ≤ t, then the minimum distance of any word in C fromthe all zeros codeword is at least t + 1. Since C is linear this implies that the minimum distance of C is
at least t + 1.


Pt
t−1
n −mp(1−p)g−1cc
If Elowis the complement of Elow , then P (Elow) ≤≤ t2 nt e−mp(1−p)
g=1 g g etc
, we get that P(Elow) ≤ e−∆0 where ∆0 := mp(1−p)t−1 −t log nefor t < n2 . Using nt ≤ nett −2 log t.
t−1
Since m = n and p and t are constants, we have that ∆0 ≥ m≥ 4C · n for all n large and2 p(1 − p)some constant C > 0.
Case II (t + 1 ≤ #S ≤ δn): For a right vertex bj , we recall that Rj is the random set of (left)
neighbours of the vertex bj . Define the event Fj (S) := {# (Rj ∩ S) is odd}. If FS
j (S) occurs, then theword v(S) would cause a parity check violation at the right vertex bj . Therefore if 1≤j≤m Fj (S) occurs,T Sthen v(S) ∈/ C. Define the event Eup := S
F
(S)
where the intersection is with respect toj1≤j≤m
all sets S whose cardinality lies between t + 1 and δn. Extending the above argument we see that if Eup
occurs, then there is no word in C whose distance from the all zeros codeword lies between t + 1 and δn.
Combining, we have that if Elow ∩ Eup occurs, then the minimum distance of the code is at least δn + 1.
For any right vertex bj , the number of left neighbours #Rj is Binomially distributed with parameters n and p. Therefore for a deterministic set S with #S = g, the cardinality ofj ∩S) set #(Rg−kPthe randomg k
is Binomially distributed with parameters g and p. Therefore P(Fjc (S)) ==0≤k≤gk p (1 − p)k even1
1
g
2 + 2 (1 − 2p) . Let 0< η < 12 be a small constant.
Using g ≥ t + 1 and choosing t sufficiently
T

m
m
1
1
1
c
c
large, we then get that P Fj (S) ≤ 21−η and so P
= 2(1−η)n. Further, there
j=1 Fj (S) ≤ 21−ηnare g sets of cardinality g and so using standard Hamming ball estimates (Proposition 3.3.1 [2]), we Pδnn11c≤· 2(1−η)n≤ 2βn, where β := (1 − η) − H(δ). Since  > H(δ) strictly, we
get that P Eup
g=t+1 gchoose η > 0 small enough so that β is strictly positive. Fixing such an η, we define Edist := Elow ∩ Eup
1≥ 1 − e−3Cn for all n large, where the constant C > 0 is as inand get that P(Edist ) ≥ 1 − e−4Cn − 2βnthe estimate for ∆0 in Case I above.
Finally, to incorporate the constraints, we use the fact that En occurs with probability at least e−Cn
for all n large and so P (En ∩ Edist ) ≥ e−Cn − e−3Cn > 0. This implies that there exists an n−lengthlinear code with relative distance at least δ, redundancy at most , and satisfying the constraint En .
Acknowledgments. I thank IMSc faculty for crucial comments and also thank IMSc for my fellowships.

References[1] M. Sipser and D. Spielman, Expander codes, IEEE Transactions on Information Theory, 42 (1996), 1710–
1722.
[2] V. Guruswami,
A. Rudra and M. Sudan,
Essential Coding Theory.
https://cse.buffalo.edu/faculty/atri/courses/coding-theory/book/web-coding-book.pdf, 2019.

Link:[3] B. Hemenway, R. Ostrovsky and M. Wootters, Local correctability of expander codes, Information andComputation, 243 (2015), 178–190.

33

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Random Derangement with Fixed Number of Cycles
Nattawut Phetmak1
1

Department of Computer Engineering, Faculty of Engineering, Kasetsart University, Bangkok 10900, Thailand
nattawut.p@ku.th (N. Phetmak)

Abstract
We study the uniform sampling of permutations without fixed points, or derangements, with mcycles. Since the number of cycles in a random derangement tends to the standard distribution, using
rejection sampling to randomly select derangements with the desired number of cycles can take along time. We propose a method for generating a uniformly random n-derangement with m cycles
in time O(n3 ) using dynamic programming.
Keywords: derangement, random generation, dynamic programming, polynomial time algorithm.
2010 MSC: Primary 05A05; Secondary 68R05, 90C39, 11B73.

1Introduction
12 ··· nA derangement is a permutation without fixed points. A notation π = π(1)π(2) ··· π(n) denotes the
permutation; thus, a derangement requires ∀i[i 6= π(i)]. Also, n-permutation and n-derangement denotea permutation and a derangement over a set of n items respectively. Pierre Remond de Montmort
proposed a problem for counting the number of distinct derangements in the first edition of his essay,then proved the result in the second edition, which Nicolaus I Bernoulli also P
derived about the samentime [1]. The result size of n-derangements, in its simplified form, is d(n) = n! i=0 (−1)i /i!, using theinclusion-exclusion principle.
There are d(n)/n! ≈ 1/e chance that a random permutation will become a derangement. Thus,
arises a simple rejection sampling algorithm that has an expectation of O(n) number of randoms untila derangement is found. Recently, there have been efforts to efficiently generate a random derangement
by reducing the number of tries [3–5].
A permutation may be viewed as a set of disjoint cycles, where each cycle is a list of elements after
repeatedly applying the permutation until arriving at the initial element. A cycle of length k may be
written as a k-cycle. For example, the permutation π = ( 11 23 32 46 54 65 ) has three cycles. The first cycleis (1), which is 1-cycle, or a fixed point. Next is (2, 3), a 2-cycle, also called a transposition. Lastly,(6, 5, 4), in that order, is a 3-cycle.
Sattolo also gave an algorithm for generating a cyclic permutation, an n-derangement with exactly
one n-cycle [7]. The number of such permutations are only (n − 1)!, however.
A Stirling number of the first kind, denoted by s(n, m), counts the number of distinct n-permutations
with m cycles [6]. Besides, Flajolet and Soria have shown that the distribution of the number of cycles
of derangements tends to the standard distribution [2]. Consequently a rejection sampling technique forfinding derangement of a specific number of cycles is made difficult.
Employing a dynamic programming technique, we propose a method for generating a uniformly
random n-derangement with m cycles in O(n3 ) time.

2

Method and Result

The outline for our algorithm works as follow, given n items and m cycles, we compute a 3-dimensional
table. Then we uniformly sample an index i over possible number of distinct n-derangement with mcycles. We use the index i to reconstruct the type of the permutation first, then find the index cyclic
permutation for cycle of each length.
34

N. Phetmak

n-derangement with m cycles

TJCDCGGG2021

Let type t of an n-permutation be an n-tuple of integers such that for 1 ≤ b ≤ n, tb counts the numberof b-cycle of the permutation. For example, π = ( 12 24 31 43 55 68 77 86 ) has two fixed points, one transposition,
and a 4-cycle. Thus, the type of the permutation is t = (2, 1, 0, 1, 0, 0, 0, 0). To simplify the notation, we
may think of a type as having infinite trailing zero’s.QAdditionof types is simply vector addition. The
number of distinct n-permutations of type t are n!/ b btb (tb )! .
We use a function T (n, m) to compute a set of distinct types of n-derangement with m cycles, which
can be described by this recurrence relation
n-tuplenz}|{o
T (n, 1) = (0, · · · , 0, 0, 1) ,jnkon
, t ∈ T (n − b, m − 1), a < b, ta = 0 .
T (n, m) = (0, · · · , 0, 0, 1) + t 2 ≤ b ≤|{z}mb-tupleUsing dynamic programming approach, we evaluate the recurrence into a 3-dimensional table A,
where A(n, m, r) counts distinct n-derangements with m cycles where the length of each cycle is no lessthan r. Each entry in A can be computed in O(1) time. Note that r ≤ n; thus, O(n2 m) time is required
to compute the table.
To find a random n-derangement of m cycles, we uniformly sample an index 1 ≤ i ≤ A(n, m, 2).
Next, we reconstruct the type of the permutation corresponding to the index i by backtracking the tablein n steps. For each step b we considering every possibility of using any number of b-cycles. Thus, thecomplexity is O(n3 ) time for type reconstruction. Finally, we split the index i into subindexes, and findthe indexed permutation for each cycle in the reconstructed type, which, for each set of b-cycle, can bedone in O(b2 ) time. Hence, we obtain an algorithm with overall O(n3 ) time.
Theorem 2.1. There exists an algorithm to generate a uniform random n-derangement with m cycles
that has running time complexity O(n3 ).
Acknowledgments. The author would like to thank Jittat Fakcharoenphol for discussions on the topic.

References[1] Pierre Rémond de Montmort, Essay [sic] d’analyse sur les jeux de hazard, Jacques Quillau, 1713.
[2] Philippe Flajolet and Michele Soria, Gaussian limiting distributions for the number of components in combinatorial structures, Journal of Combinatorial Theory, Series A 53 (1990), no. 2, 165–182.
[3] Conrado Martı́nez, Alois Panholzer, and Helmut Prodinger, Generating random derangements, 2008 Proceedings of the Fifth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), SIAM, 2008, pp. 234–
240.
[4] J Ricardo G Mendonça, Efficient generation of random derangements with the expected distribution of cycle
lengths, Computational and Applied Mathematics 39 (2020), no. 3, 1–15.
[5] Kenji Mikawa and Ken Tanaka, Linear-time generation of uniform random derangements encoded in cycle
notation, Discrete Applied Mathematics 217 (2017), 722–728.
[6] John Riordan, An introduction to combinatorial analysis, Princeton University Press, 2014.
[7] Sandra Sattolo, An algorithm to generate a random cyclic permutation, Information processing letters 22(1986), no. 6, 315–317.

35

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Multiple Hook Removing Game
Tomoaki Abuku1 and Masato Tada2
1

National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430
University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577 Japan
buku3416@gmail.com (Tomoaki Abuku), t-d-masato@math.tsukuba.ac.jp (Masato Tada)
2

Abstract
The Hook Removing Game is a combinatorial game with positions represented by Young diagrams. It was invented in 1970 by Mikio Sato, who also found a formula for the G-values∗ [2, 3].
Young diagrams also correspond to symmetric groups. More generally, there are several companion
diagrams to Young diagrams, called d-complete posets, to which some Weyl groups correspond. In
2001, Noriaki Kawanaka showed that the Hook Removing Game using d-complete posets, called thePlain Game, can be written as a closed expression of the G-values [1].
In this study, we investigated the properties of hooks in numbered rectangular Young diagramsand the G-values of positions represented by such diagrams in the Multiple Hook Removing Game† .
In particular, we obtained closed expressions for the G-values in the case of the diagrams whose
starting position is 1 × n, 2 × n, n × n, and n × (n + 1), respectively. Moreover, we have clariﬁed therelationship between the G-values in the case of the diagrams whose starting position is m × n andm × (n + 1).
Keywords: Combinatorial Game, Hook Removing Game, G-value, Young Diagram, Weyl Group.
2010 MSC: Primary 91A46; Secondary 05E10.

1Introduction

Let Ym,n be a numbered rectangular Young diagram‡ with m rows and n columns.
Deﬁnition 1.1. For box (i, j) of Young diagram Y ,hY (i,j) := {(i, j)} ⊔ {(i′ , j) ∈ Y | i′ > i} ⊔ {(i, j ′ ) ∈ Y | j ′ > j}is called the hook corresponding to the box (i, j).
The rules of the Multiple Hook Removing Game are as follows:
1. Given a numbered rectangular Young diagram, the two players alternate between the following
operations:(a) Select a box and remove the corresponding hook. Let C(h) be a multiset consisting of thenumbers of the removed hook.
(b) After that, if there is another hook that has the same number as C(h), remove that hook. As
long as such a hook exists, repeat this operation.
2. The last player to remove a hook wins.
Example 1.2. A and B are players, and A has the ﬁrst move.
∗ The G-value is a value deﬁned for a position in the game, and this value can be used to determine whether the ﬁrst
player has a winning strategy or the second player has a winning strategy (See [4]).
† Based on Tada’s work on a representation method using d-complete posets with special coloring for Weyl groups.
‡ For 1 ≤ i ≤ m, 1 ≤ j ≤ n, we deﬁne the number of box (i, j) as min(j − i + m, i − j + n).

36

T. Abuku et al.

3 4 3 2 1
2 3 4 3 2
1 2 3 4 3Multiple Hook Removing GameA−→

3 4 3 2 1
2 3 4 3
1 2 3

B

−
→

3 4 3 2 1
2 3

TJCDCGGG2021

B

−
→

3
2

A

−
→ ∅.

At player B’s turn, two hooks have been removed according to the game rule 1.(b).
Lemma 1.3. In the Multiple Hook Removing Game, a player can only remove a hook at most 2 times.

2

Main Results

2.1

In the case of Ym,n and Ym,n+1

Theorem 2.1. If m+n is even and m ≤ n, then the G-value of a position in the Multiple Hook Removing
Game whose starting position is Ym,n and the G-value of a position in the Multiple Hook Removing Game
whose starting position is Ym,n+1 are equal.

2.2

In the case of Y1,nTheorem 2.2. In the Multiple Hook Removing Game, the G-value of the starting position Y1,n (i.e.,G(Y1,n )) is as follows:{n(n is odd),
G(Y1,n ) =n − 1 (n is even).

2.3

In the case of Y2,nTheorem 2.3. Let Z>0 be the set of all positive integers. In the Multiple Hook Removing Game, theG-value of the starting position Y2,n (i.e., G(Y2,n )) is as follows:
3 (n = 2, 3),
G(Y2,n ) = 2 (n = 2 + 8m, 3 + 8m (m ∈ Z>0 )),
1 (otherwise).

2.4

In the case of Yn,n and Yn,n+1

Theorem 2.4. The Multiple Hook Removing Game in the numbered rectangular Young diagram Yn,n+1
and the Hook Removing Game in the shifted Young diagram (n, n − 1, n − 2, . . . , 2, 1) are isomorphic (i.e.,
the game trees are equal), and the G-values are equal (even in the intermediate positions).
Corollary 2.5. Let Yn,n or Yn,n+1 be the position in the Multiple Hook Removing Game and denote it
by Z. Then we have
G(Z) =
where

⊕

⊕

k,

1≤k≤n

denotes the sum of all numbers in binary form without carrying§ .

Acknowledgments. The authors would like to thank Yuki Motegi for his advice.

References[1] N. Kawanaka, Sato-Welter game and Kac-Moody Lie algebras, RIMS Kôkyûroku 1190 (2001), 95–106.
[2] M. Sato (Notes by H. Enomoto), On Maya game, Sûgaku-no-Ayumi 15 − 1 (Special Issue : Mikio Sato),(1970), 73-84. (in Japanese)[3] M. Sato (Notes by T. Umeda), Lectures (on Soliton Theory) at Kyoto University, RIMS Kyoto University,
1989. (in Japanese)[4] A. N. Siegel, Combinatorial Game Theory, PIRT Conference American Mathematical Society (2013).
§ This notion is equivalent to “bitwise XOR”.

37

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Complete Geometric Graphs with no Partition into Plane
Spanning Trees
Johannes Obenaus1 , and Joachim Orthaber2
1

Department of Computer Science, Freie Universität Berlin, Germany
Institute for Software Technology, Graz University of Technology, Austria
johannes.obenaus@fu-berlin.de, joachim.orthaber@student.tugraz.at
2

Abstract
It is a long-standing conjecture that every complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Despite several approaches, it remained open to this day. In
this note we provide a family of counterexamples, based on bumpy wheel sets.
Keywords: plane spanning trees, bumpy wheel sets.
2010 MSC: Primary 52C15; Secondary 05C05, 05C10.

1Introduction

A geometric graph G = (P, E) consists of a set of vertices P , which are points in the plane in general
position, and a set of straight-line edges E. Hurtado et al. [4] introduced the famous and long-standing
conjecture that the edges of any complete geometric graph on 2n vertices can be partitioned into n plane
spanning trees (see also [1]). Despite a lot of effort, very little is known about this problem, namely only
for P being in convex position [2] or in regular wheel configuration [6] an affirmative answer is known.
Also, many closely related problems have been studied, e.g. in the context of packing plane spanning
trees into complete graphs: Biniaz and Garcı́a [3] showed that ⌊n/3⌋ plane spanning trees can always be
packed into a complete geometric graph.
In this note, we provide a family of counterexamples to the conjecture above. The point sets weare using for our construction, so-called bumpy wheel sets, have been introduced in [5]. For positive
parameters k and ℓ, a set P of k · ℓ + 1 points is in bumpy wheel configuration if k · ℓ of its points areplaced along a circle into k groups each containing ℓ points and the remaining point being the center ofthe circle. The points within each group are ε-close (see Figure 1 for an illustration). We denote thecorresponding complete geometric graph by BWk,ℓ . Since we are only concerned with graphs having aneven number of vertices, we consider BWk,ℓ only for odd parameters k and ℓ. For BW3,5 and BW3,7 weshow the following even stronger statement:
Theorem 1.1. The edges of BW3,5 cannot be partitioned into n2 = 8 color classes such that each color
class is plane, contains exactly n − 1 = 15 edges, and is triangle free. BW3,7 can’t even be partitioned
into n2 plane subgraphs.
In this note we will present a computer assisted proof based on an ILP formulation. On the other
hand, we omit a pen and paper proof that BWk,ℓ cannot be partitioned into plane spanning trees for any
odd k ≥ 3 and ℓ ≥ 5 (which is based on structural insights) from this version due to space constraints.

2

The ILP ModelIn this section we give an overview of the computer assisted proof of Theorem 1.1. Given a geometricgraph G = (P, E) and a fixed number m of available colors as input, our ILP contains a binary variable xe,c for each edge-color combination, i.e. in our setting there are n2 · m variables. xe,c being 1 thencorresponds to edge e receiving color c.
1 supported by ERC StG 757609

38

J. Obenaus et al.

Edge Partitions of Kn

TJCDCGGG2021

Figure 1: Partition of the bumpy wheel BW3,5 into 8 plane subgraphs. A partition into plane spanning
trees is not possible.
We need to implement the following constraints, enforcing that every edge receives exactly one color,crossing edges receive different colors, ensuring n − 1 edges in each color class, and forbidding monochromatic triangles:mX

xe,c = 1∀e ∈ E(2.1)∀c ∈ {1, . . . , m}; ∀e, f crossing(2.2)∀c ∈ {1, . . . , m}

(2.3)

for each triangle e, f, g; ∀c ∈ {1, . . . , m}

(2.4)

c=1

xe,c + xf,c ≤ 1
Xxe,c = n − 1e∈Exe,c + xf,c + xg,c ≤ 2

All experiments were run on an Intel Core i5, 1.6 GHz, 16 GB RAM running macOS Big Sur Version 11.4. All algorithms were implemented in Python 3.9.1, and for solving the ILP we used Gurobi
Optimizer Version 9.1.2 with default settings.
For BW3,5 and m = 8 as input, our program reports an infeasible ILP (taking less than a minute).
Furthermore, for BW3,7 and m = 11 as input our program reports an infeasible ILP even if omitting theconstraints (2.3) and (2.4) (taking roughly 5h). Figure 1 shows a partition of BW3,5 into plane subgraphs
found by the program, if omiting the triangle constraint (2.4).

3

OutlookA natural generalization would be to partition complete geometric graphs into k-planar or k-quasiplanar
subgraphs, which is part of our currently ongoing work on this project.
Acknowledgments. We are particularly grateful to Patrick Schnider for suggesting the bumpy wheel
sets. We also thank the organizers and participants of the 5th DACH Workshop on Arrangements andDrawings, that took place in March 2021 (online) and was funded by Deutsche Forschungsgemeinschaft(DFG), the Austrian Science Fund (FWF) and the Swiss National Science Foundation (SNSF).

References[1] http://www.openproblemgarden.org/op/partition_of_complete_geometric_graph_into_plane_trees.
[2] Frank Bernhart and Paul C Kainen, The book thickness of a graph, Journal of Combinatorial Theory, Series
B 27 (1979), no. 3, 320–331.
[3] Ahmad Biniaz and Alfredo Garcı́a, Packing plane spanning trees into a point set, CoRR abs/1803.02385(2018).
[4] Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, and David R. Wood, Partitions of complete geometric graphs into plane trees, Graph Drawing (Berlin, Heidelberg) (János Pach, ed.), Springer Berlin Heidelberg,
2005, pp. 71–81.
[5] Patrick Schnider, Partitions and packings of complete geometric graphs with plane spanning double stars andpaths, Master’s thesis, ETH Zürich, 2015.
[6] Hazim Michman Trao, Gek L Chia, Niran Abbas Ali, and Adem Kilicman, On edge-partitioning of complete
geometric graphs into plane trees, arXiv preprint arXiv:1906.05598 (2019).

39

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

On the Size of Minimal Separators for
Treedepth Decomposition
Zijian Xu1 , Vorapong Suppakitpaisarn2
1

Preferred Networks Tokyo, JapanThe University of Tokyo, Tokyo, Japan
joe@preferred.jp (Z. Xu), vorapong@is.u-tokyo.ac.jp (V. Suppakitpaisarn)
2

Abstract
The treedepth of a graph is a parameter that describes how it is close to a star graph. Recently,
many parameterized algorithms have been developed for solving various problems on graphs withsmall treedepth. Since those algorithms are based on treedepth decompositions of input graphs, many
solvers are developed to find optimal treedepth decompositions. Many practical solvers enumerate
minimal separators to construct a treedepth decomposition in a top-down way. The bottleneckof these algorithms is often the enumeration of minimal separators. Therefore, to speed up thecomputation, it is essential to reduce the number of minimal separator candidates that are used foran optimal solution. We show that, to obtain an optimal treedepth decomposition, it is sufficient toconsider only minimal separators of size at most 2tw, where tw denotes the treewidth of the input
graph. We also show that this upper bound is tight up to a constant.
Keywords: Graph Algorithms, Treedepth, Minimal Vertex Separator, Treewidth.
2010 MSC: Primary 68R10; Secondary 05C85, 05C40, 68W40.

1Introduction

Treedepth [9] is a graph parameter that has recently received attentions [7, 8]. Recently, treedepth
has been well studied for parameterized algorithms. Given a treedepth decomposition of height k,
Iwata et al. [8] gave an O(km) algorithm for Maximum Matching and O(k(m + n log n)) algorithms
for several problems including Weighted Matching and Negative Cycle Detection. Hegerfeld
and Kratsch [7] developed single-exponential FPT algorithms for connectivity problems including Connected Vertex Cover and Steiner Tree by extending the Cut&Count technique for treewidth by
Fomin et al. [5].
To make use of those parameterized algorithms, we need a treedepth decomposition of a small height.
The treedepth is defined as the minimum height of all treedepth decompositions. Computing treedepth
is NP-hard even for chordal graphs [3]. However, the appearance of the formula is simple, and it has
been re-discovered as different parameters in many fields. Denote the set of connected components of agraph G by C(G). The treedepth of a graph G, denoted as td(G), can be computed as follows [9]:
max td(G0 )(if G is not connected)
G0 ∈C(G)td(G) = 1(|V | = 1)
1 + min td(G\v) (otherwise),

(1.1)

v∈V

where G\v denotes the graph obtained from G by deleting vertex v. The naive dynamic programming
based on Equation (1.1) yields an algorithm of O∗ (2n ), where O∗ notation denotes the complexity hiding
the polynomial factor. Speeding up the computation of treedepth is essential for those parameterizedalgorithms. Fomin et al. [4] modified the dynamic programming of Equation (1.1) and obtained an exact
exponential algorithm of O∗ (1.9602n ) time. Reidl et al. [10] proposed a parameterized algorithm which
2
decides whether the treedepth is at most k in 2O(k ) · n time.
40

Z. Xu et al.

2

On the Size of Minimal Separators for Treedepth Decomposition

TJCDCGGG2021

Top-down Algorithm and its Bottleneck

In addition to the algorithms with theoretical guarantee mentioned in the previous paragraph, thealgorithms that work practically fast for real-world graphs are also studied. Two major approaches used
for the practical algorithms were the top-down algorithm by recursively choosing minimal separators (e.g.
[1]) and the bottom-up positive-instance dynamic programming (e.g. [11]). In the top-down algorithm,
the computation is based on the following formula [2]:
max td(G0 )(if G is not connected)
G0 ∈C(G)td(G) = |V |(2.1)
(if G is complete)
|S| + min td(G\S) (otherwise),S∈∆

where ∆ is the set of all minimal separators of G and G\S denotes the graph obtained from G by
deleting vertex set S. Note that when G is complete, ∆ = ∅. For a connected, non-complete graph
G, a minimal separator S is called optimal top separator if td(G) = |S| + td(G\S). The bottleneck of
the computation by Equation 2.1 is the enumeration of minimal separator (i.e., the computation of ∆),because the number of minimal separators is usually quite large and it can be exponential [6]. To speed
up the algorithm, it is promising to reduce the number of candidate minimal separators that can be an
optimal top separator. Experimentally, it is known that the size of some optimal top separator can bevery small. In order to design a practically fast exact algorithm for treedepth, it has been an important
challenge to analyze the size of optimal top separators.

3

Main Results

In this paper, we show that we can obtain treedepth decomposition by considering only minimal separators of size at most 2tw, where tw denotes the treewidth of the input graph. We can replace the set∆ at Equation 2.1 by the set ∆0 = {S ∈ ∆ | |S| ≤ 2tw} and still obtain td(G). Moreover, we show that
this upper bound is tight up to a constant.

References[1] Ruben Brokkelkamp, Raymond van Venetië, Mees de Vries, and Jan Westerdiep, PACE Solver Description:
tdULL, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020), vol. 180,
Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2020, pp. 29:1–29:4.
[2] Jitender S Deogun, Ton Kloks, Dieter Kratsch, and Haiko Müller, On the vertex ranking problem for trapezoid, circular-arc and other graphs, Discrete Applied Mathematics 98 (1999), no. 1-2, 39–63.
[3] Dariusz Dereniowski and Adam Nadolski, Vertex rankings of chordal graphs and weighted trees, Information
Processing Letters 98 (2006), no. 3, 96–100.
[4] Fedor V Fomin, Archontia C Giannopoulou, and Michal Pilipczuk, Computing tree-depth faster than 2n ,Algorithmica 73 (2015), no. 1, 202–216.
[5] Fedor V Fomin, Daniel Lokshtanov, Saket Saurabh, Michal Pilipczuk, and Marcin Wrochna, Fully
polynomial-time parameterized computations for graphs and matrices of low treewidth, ACM Transactions
on Algorithms (TALG) 14 (2018), no. 3, 1–45.
[6] Serge Gaspers and Simon Mackenzie, On the number of minimal separators in graphs, International Workshop on Graph-Theoretic Concepts in Computer Science, Springer, 2015, pp. 116–121.
[7] Falko Hegerfeld and Stefan Kratsch, Solving connectivity problems parameterized by treedepth in singleexponential time and polynomial space, 37th International Symposium on Theoretical Aspects of Computer
Science (STACS 2020), vol. 154, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2020, pp. 29:1–29:16.
[8] Yoichi Iwata, Tomoaki Ogasawara, and Naoto Ohsaka, On the Power of Tree-Depth for Fully Polynomial
FPT Algorithms, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018), vol. 96,
Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2018, pp. 41:1–41:14.
[9] A. Pothen, The complexity of optimal elimination trees, Tech. report, Pennsylvannia State University Technical Report CS-88-13, 1988.
[10] Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar, A faster parameterizedalgorithm for treedepth, International Colloquium on Automata, Languages, and Programming, Springer,
2014, pp. 931–942.
[11] James Trimble, PACE Solver Description: Bute-Plus: A Bottom-Up Exact Solver for Treedepth, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020), vol. 180, Schloss Dagstuhl–
Leibniz-Zentrum für Informatik, 2020, pp. 34:1–34:4.

41

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Hamiltonian Decompositions of Complete 4-partite 3-uniform
Hypergraphs
Chutima Saengchampa1 and Chariya Uiyyasathian1
1

Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, PhayathaiRoad, Pathumwan, Bangkok 10330, Thailand
chutima.scp@gmail.com (C. Saengchampa), chariya.u@chula.ac.th (C. Uiyyasathian)Abstract
Based on the definition of Hamiltonian cycles given by Katona and Kierstead, we provide a(3)
construction of Hamiltonian decompositions of complete 4-partite 3-uniform hypergraphs K4(2m) ,where 2m is the size of each partite set.
Keywords: Hamiltonian decomposition, uniform hypergraph, multipartite hypergraph.
2010 MSC: Primary 05C65.

1Introduction

A Hamiltonian decomposition of a hypergraph is a partition of its hyperedge set into mutually disjoint
Hamiltonian cycles. The definition of a Hamiltonian cycle can be extended to hypergraphs in various
ways. The definition in this paper based on a Hamitonicity of cycles for k-uniform hypergraphs H(V, E)
of order n given by Katona and Kierstead [5] : a Hamiltonian cycle of H is a cyclic ordering C = (v1 v2. . . vn ) of all n elements of V such that k consecutive vertices form a hyperedge in E.
(3)
The existence problem of a Hamiltonian decomposition for the complete 3-uniform hypergraph Kn
has been studied by many authors with a variety of construction techniques. However, the result wasestablished only for some n where n ≤ 46 or n = 2m for m ≥ 2 (see [1, 3, 6, 7]). The existence problem
was also studied for complete multipartite k-uniform hypergraphs, defined as follows :
(k)

(k)

Definition 1.1. A complete multipartite k-uniform hypergraph Kn1 ,n2 ,...,nt or Kn1 ,n2 ,...,nt (V1 , V2 ,. . ., Vt )is a hypergraph with vertex set V = V1 ∪ V2 ∪· · · ∪ Vt where |Vi | = ni for all i ∈ {1, 2, . . . , t}, andE(H) = {e : e ⊂ V, |e| = k and |e ∩ Vi | < k for i ∈ {1, 2, . . . , t}}.
(k)

(k)

In particular, if ni = n for all i ∈ {1, 2, . . . , t}, then Kn, n, . . . n is denoted by Kt(n) .
| {z }
t(k)

In literature, the problem of Hamiltonian decompositions of Kt(n) has been investigated only for the(3)case k = 3 and t = 2 and 3. Wang and Jirimutu [4] studied on Kn,n when n is a prime number in 2001.
Later on, Xu and Wang [7] provided a complete study for all n ≥ 2 in 2002. The study for complete(3)
tripartite 3-uniform hypergraphs, Kn,n,n , was also completed by Boonklurb et al. [2] in 2015. Continuingalong this line, we are interested in the case t = 4 and n is any even positive integer. In other words, we(3)
provide a construction of Hamiltonian decompositions of K4(2m) for all positive integer m.

2Main Results
(3)

(3)

We first classify hyperedges of K4(2m) into two types. Let e be a hyperedge of K4(2m) , if e contains at
most one vertex from each partite set, e is then called a hyperedge of Type 1, otherwise (that is e contains(3)
two vertices from a partite set) e is called a hyperedge of Type 2. Let Ti (K4(2m) ) be the subhypergraph(3)of K4(2m) which consists of all hyperedges of Type i for i ∈ {1, 2}.
42

C. Saengchampa et al.

Hamiltonian Decompositions of 4-partite 3-uniform Hypergraphs

TJCDCGGG2021(3)

Let Vi = {ai1 , ai2 , . . . , ai2m } for i ∈ {1, 2, 3, 4} be partite sets of K4(2m) (V1 , V2 , V3 , V4 ). We will give a
(3)

(3)

(3)

construction of K4(2m) by decomposing T1 (K4(2m) ) and T2 (K4(2m) ) into Hamiltonian cycles separately.
In our construction, we write cycle C as (P1 P2 . . . Ps ) if vertices along the cycle C are partitioned into
paths Pj (a sequence of vertices) along this cycle.
(3)

To decompose T1 (K4(2m) ), we establish a stronger result by providing a Hamiltonian decomposition(3)of the subhypergraph T1 (K4(n) ) for all positive integers n. The construction uses a 1-factorization F ofthe complete bipartite graph Kn,n .
(3)Theorem 2.1. The hypergraph T1 (K4(n) ) has a Hamiltonian decomposition for all positive integer n.
(3)

On the other hand, to decompose T2 (K4(2m) ), we use a 1-factorization F of K2m on the vertex
set {1, 2, . . . , 2m} and a collection D of three permutations of {1, 2, 3, 4} with a certain property. We
then construct a collection of Hamiltonian cycles, each of which is composed of two paths of order 4m,
C = {Ct (D, F ) = (P1t P2t ) : t ∈ {0, 1, . . . , n − 1}, D ∈ D, F ∈ F }. The construction is separated into twocases depending on the parity of m.
(3)

Theorem 2.2. The subhypergraph T2 (K4(2m) ) has a Hamiltonian decomposition when m is odd.
(3)

Theorem 2.3. The subhypergraph T2 (K4(2m) ) has a Hamiltonian decomposition when m is even.
(3)

Example 2.4. An illustration of C0 (D, F ) which are in the construction for T2 (K4(2m) ) when m = 5,
D = (1, 3, 4, 2) and F = {{j, f (j)} : j ∈ {1, 2, 3, 4, 5}} is a 1-factor of K10 (the vertex set of K10 isrelabeled as {1, 2, 3, 4, 5, f (1), f (2), f (3), f (4)f (5)}).
In Figure 1, for x ∈ {1, 2, 3, 4}, each vertex axℓ in the partite set Vx in the cycle C0 (D, F ) is represented
by its subscript ℓ. Moreover, the solid lines indicate two consecutive vertices in the same path, and thedash lines indicate two consecutive vertices from different paths.
..

1

f (1)

f (5)

5

1

f (1)

f (5)

5

2

f (2)

f (4)

4

2

f (2)

f (4)

4

3

f (3)

f (3)

3

3

f (3)

f (3)

3

4

f (4)

f (2)

2

4

f (4)

f (2)

2

5

f (5)

f (1)

1

5

f (5)

f (1)

1

V1

V3

V4

V2

(3)

Figure 1: C0 (D, F ) of T2 (K4(10) ).
Although the construction in Theorem 2.3 is a bit more complicated, the proof of the Hamiltonicity
of cycles and the decomposition of C uses a similar approach as in Theorem 2.2.
Acknowledgments. The first author would like to thank Science Achievement Scholarship of Thailand
for her financial support.

References[1] R. F. Bailey and B. Stevens, Hamiltonian decompositions of complete k-uniform hypergraphs, Discrete Math.
310 (2010), 3088–3095.
[2] R. Boonklurb and S. Singhun and S. Termtanasombat, Hamiltonian decomposition of complete tripartite
3-uniform hypergraphs, East-West J. Math. 17 (2015), 48–60.
[3] H. Huo and L. Qi Zhao and W. Feng and Y. Sheng Yang and Jirimutu, Decomposing complete 3-uniform
hypergraph into Hamiltonian cycles, Acta Math. Sinica (Chin. Ser.) 58 (2015), 965–976.
[4] Jirimutu and J. Wang, Hamiltonian decomposition of complete bipartite r-hypergraphs, Acta Math. Appl.
Sin. Engl. Ser. 17 (2001), 563–566.
[5] G. Y. Katona and H. Kierstead, Hamiltonian chains in hypergraphs, J. Graph Theory 30 (1999), 205–212.
[6] M. Meszka and A. Rosa, Decomposing complete 3-uniform hypergraph into Hamiltonian cycles, Australas.
J. Combin. 45 (2009), 230–253.
[7] B. Xu and J. Wang, On the Hamiltonian cycle decompositions of complete 3-uniform hypergraphs, Electron.
Notes Discrete Math. 328 (2002), 722–733.

43

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

On the Edge Irregularity Strength of Some Disjoint Union
GraphsRismawati Ramdani1 , Hasni Rahmani Rohim2 , and Irsa Islammeidini Rusviantir1
1

Department of Mathematics, Faculty of Science and Technology, UIN SUnan Gunung Djati Bandung,Indonesia
2
Magister of Mathematics Education, Universitas Pasundan, Bandung, Indonesia
rismawatiramdani@uinsgd.ac.id (R. Ramdani), hasnirahmanirohim@gmail.com (H. R. Rohim),
Irsaislammeidini@gmail.com (I. Rusviantir)
Abstract
Let G = (V (G), E(G)) be a graph and k be a positive integer. A vertex k-labeling f : V (G) →{1, 2, . . . , k} is called an edge irregular labeling if there are no two edges with the same weight, where
the weight of an edge uv is f (u) + f (v). The edge irregularity strength of G, denoted by es(G), is theminimum k such that G has an edge irregular k-labeling. This labelings were introduced by Ahmad,
Al-Mushayt, and Bača in 2014. In this paper, we determine the edge irregularity strength of disjoin
union of stars, disjoint union of cycles, and disjoint union of paths.
Keywords: cycle, disjoint union edge irregular labeling, edge irregularity strength, path, star.
2010 MSC: 05C78.

1Introduction

Graph labeling is a topic in graphs that is growing quite rapidly. Thehe are different types of graphlabeling. One of the type of graph labeling is edge irregular labeling. An edge irregular labeling is a kindof vertex labeling, which is a labeling that label only vertices in a graph. The edge irregular labelingwas introduced by Ahmad et al. [1] In 2014. The definition of edge irregular labeling is given the thedefinition below.
Definition 1.1. For an integer k, a total labeling f : V (G) → {1, 2, . . . , k} is called an edge irregulark-labeling of G if every two distinct edges e1 and e2 in E satisfy wf (e1 ) 6= wf (e2 ), where wf (e1 = uv) =f (u) + f (v).
Definition 1.2. The minimum k for which a graph G has an edge irregular k-labeling, denoted byes(G), is called the edge irregularity strength of G.

2

Preliminaries

In a paper published in 2014, Martin Baca and Ali Ahmad got a lower bound of the edge irregularity
strength of arbitrary graph as follows.
Theorem 2.1. Theorem 1. [1] Let G = (V, E) be a graph with the maximum degree ∆, then|E(G)| + 1
es(G) ≥ max, ∆(G)
2.
In 2020, Ahmad et al performed a computer based experiment dealing with the edge irregularity
strength of complete bipartite graphs [2]. In 2017, Ahmad et al gave the edge irregularity strengths ofsome chain graphs and the join of two graphs. They also introduced a conjecture and open problems forresearchers for further research. [3] Besides that, the edge irregularity strength of graphs are given by44

R. Ramdani et al.

On the Edge Irregularity Strength of Some Disjoint Union Graphs

TJCDCGGG2021

Imran et al. in [4]. In the paper, they determined the egde irregularity strength of caterpillars, n-stargraphs, kite graphs, cycle chains and friendship graphs. In [5], Tarawneh et al determined he edge irregularity strength of corona product of cycle with isolated vertices. Furthermore, In [6], Tarawneh et al
gave the exact value of edge irregularity strength for triangular grid graph , zigzag graph and Cartesian
product of Pn , Pm , and P2 .

3Main Results

Theorem 3.1. Let Sn be star with n + 1 vertices and mSn be m copies of Sn . Then for m ≥ 1 andn ≥ 1,

mn + 1.
es(mSn ) =2
Theorem 3.2. Let Pn be path with n vertices and mPn be m copies of Pn . Then for m ≥ 2 and n ≥ 2,
mn − m + 1
es(mPn ) =.
2
Theorem 3.3. Let Cn be cycle with n vertices and mCn be m copies of Cn . Then for m ≥ 3 and n ≥ 3,
mn + 1es(mPn ) =.
2
Acknowledgments. Author acknowledges the support by UIN Sunan Gunung Djati Bandung. The
author wish to thank the referees for their thoughtful suggestions.

References[1] M. Bača, On irregular total labellings, Discrete Math. 307 (2017), 1378–1388.
[2] A.Ahmad, OBS. Al-Mushayt, and M.Bača, On edge irregularity strength of graphs, Applied Mathematics
and Computation. 243 (2017), 607–610.
[3] A.Ahmad, A.Guptaa. and R.Simanjuntak, Computing the edge irregularity strengths of chain graphs and thejoin of two graphs, Electronic Journal of Graph Theory and Applications. 6(1) (2018), pp 201–207.
[4] A.Ahmad, MA. Asim, B.Asshiri, and A.Semaničová-Feňovčı́ková, Computing the edge irregularity strengthof bipartite graphs and wheel related graphs, Fundamenta Informaticae. 174(1) (2020), pp 1–13.
[5] M.Imran, A.Aslam, S.Zafard, and W.Nazeere, Further results on edge irregularity strength of graphs, Indonesian Journal of Combinatorics. 1(2) (2017), 82–91.
[6] I.Tarawneh, R.Hasni, and A.Ahmad, On the edge irregularity strength of corona product of cycle with isolated
vertices, AKCE International Journal of Graphs and Combinatorics. 13 (2016), 213–217.
[7] I.Tarawneh, R.Hasni, and A.Ahmad, On the edge irregularity strength of grid graphs, AKCE International
Journal of Graphs and Combinatorics. 17 (2020), 414–418.

45

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Tiling the Plane Connectively with Wang Tiles
Chao Yang1
1

School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China
sokoban2007@163.com (C. Yang)
Abstract
Inspired by the Nintendo game Zelda, the problem of tessellating the plane by translated copies
from a set of Wang tiles is generalized to the connected tiling problem. We solve the connected tilingproblem for the case that there is only one door color and one wall color.
Keywords: Wang tile, tiling, connectivity, complexity, Zelda.
2010 MSC: Primary 52C20; Secondary 68Q17.

1Introduction

The motive of this study comes from two sources. The first source is original Domino Problem posed by
Wang in 1961 [1]. A Wang tile is a unit square with each edge assigned a color. Given a finite set of
Wang tiles, we would like to tile the plane with translated copies of them such that the common edgeof each pair of neighboring Wang tiles must have the same color. This is known as Wang’s Domino
Problem. This problem has been shown to be undecidable by Berger [2].
The second source is the video game Zelda published by Nintendo. One of the most successful title of
the Zelda series is The Legend of Zelda: Link’s Awakening which was released for the handheld console
GameBoy in 1993 and for GameBoy Color in 1998. In 2019, the game was remade and published for
the Switch console 26 years after its first release. Besides the huge improvement in graphics, the Switch
version is almost identical to the GameBoy version in which the player walks in different 2D dungeons
to solve puzzles. One major new features added to the Switch version is a mini-game which allow thegamers to create their own dungeons by putting together square rooms (See Figure 1).

Figure 1: Screenshots of the game Zelda
If we regard the doors of the rooms as one color, and the walls of the rooms another color, the Zelda’s
dungeon-making mini-game is just a special case of Wang’s problem. The Zelda’s dungeon-making puzzle
and Wang’s problem differ in two points: (1) the Zelda’s dungeon need not fill up the entire plane; (2)the Zelda’s dungeon must be connected, that is, the player must be able to walk from one room to any
other rooms of the dungeon through doors.
This inspires us to propose the following problem, which incorporates connectivity into the original
Wang’s problem.
46

C. Yang

Tiling Connectively

TJCDCGGG2021

Problem 1.1 (Connected Tiling Problem). Given two finite sets of colors D (the set of door colors) andW (the set of wall colors), and a finite set of Wang tiles. Each side of a Wang tile is asigned to one of
the colors from either D or W . Can we tile the whole plane by translated copies from this set of Wangtiles such that the tiling is connected (i.e. walk from one tile to another only through sides with colorsin D)?
If W = ∅, then any tiling must be connected, therefore in this case the problem is the same as theoriginal Wang’s problem. So the connected tiling problem includes the original Wang’s tiling problem
as a special case. As a result, the general connected tiling problem must also be undecidable. But
the problem can be solvable if we limit the number of colors. In this paper, we study the case that
|D| = |W | = 1. We show that in this case the problem is solvable by giving the complete list of sets ofWang tiles which can tile the plane connectively.

2

The Main Result

In this section, we only consider the connected tiling problem with |D| = |W | = 1. So the set of all
possible 16 Wang tiles are illustrated in Figure 2, with the color in D represented by blue lines.

Figure 2: Wang tiles with |D| = |W | = 1
There are 216 − 1 = 65535 different nonempty subsets of these Wang tiles. Our main result is thefollowing theorem which gives a complete classification of the subsets that can tile the plane connectively.
Theorem 2.1. For the connected tiling problem with |D| = |W | = 1, a set of Wang tiles can tile theplane connectively by translated copies if and only if it is one of the following sets:
1. sets containing the 4-door tile;
2. sets containing a pair of opposite 3-door tiles;
3. sets containing a pair of adjacent 3-door tiles, and the two straight 2-door tiles;
4. sets containing a pair of adjacent 3-door tiles, a straight 2-door tile and a complement parallel
1-door tile;
5. sets containing a pair of adjacent 3-door tiles, a complement turing 2-door tile and two complement
1-door tile;
6. sets containing one 3-door tile and two complement adjacent 2-door tiles;
7. sets containing the four turing 2-door tiles and one more straight 2-door tile;
8. sets containing the four turing 2-door tiles and one 1-door tile.
Sketch of Proof. We prove the theorem on a case-by-case study. For the cases that there exist connectedtilings, we will give detailed illustratations of the tilings. For cases without connected tilings, we willshow the impossibility by using techniques such as parity argument. Due to page limit, more definitions
and details of the proof will be provided in the conference presentation and the full paper.

References[1] H. Wang, Proving theorems by pattern recognition - II, Bell System Technical Journal 40 (1) (1961), 1–41.
[2] R. Berger, The undecidability of the domino problem, Memoirs of the American Mathematical Society 66(1966), 1–72.

47

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Generating Frames via Discretized Substitution Tilings
Luis S. Silvestre Jr.1 and Job A. Nable1
1

Department of Mathematics, Ateneo de Manila University, Quezon City 1108, Philippines
lsilvestre@ateneo.edu (L. Silvestre), jnable@ateneo.edu (J. Nable)
Abstract
This work aims to aperiodically discretize Gabor frames for L2 (Rd ). To do so, discrete sets are
obtained from aperiodic tilings of R2d called Pisot family substitution tilings, and the specific class ofprimitive substitution tiling systems. The resulting aperiodic sets namely model sets and discretizedprimitive tilings satisfying certain hole conditions generate Gabor frames for L2 (Rd ) with appropriate
window functions.
Keywords: Pisot family substitution tiling, aperiodic model set, primitive substitution tiling system, discretizedprimitive tiling, Gabor frame.
2010 MSC: Primary 52C23, 42C15; Secondary 52C22, 46B15.

1

Introduction

Gabor frames G(f, Λ) based on some window function f ∈ L2 (Rd ) and some subset Λ ⊂ R2d are essential
tools for signal analysis and quantization. Most research on Gabor frames focuses on finding appropriate
windows, or on periodic discretization via lattices. This study aims to extend such results by providing
an aperiodic discretization based on aperiodic tilings of R2d namely Pisot family substitution tilings andprimitive substitution tiling systems.

2

Preliminaries

2.1

Model Sets and Discretized Primitive Tilings

A Pisot family substitution tiling % is a primitive substitution tiling on Rd with expansion map
φ : Rd → Rd whose eigenvalues form a Pisot family [1]. If % has finite local complexity (FLC), thatis, the tiling has finitely many R-patches BR (%) up to translations, then a cut-and-project scheme
associated with % may be formulated. This produces a model set Λ(Γ) ⊂ Rd which is an aperiodic
Delone set on Rd with FLC. In this case, we say that % admits a PST model set. Define the hullΩ(Λ(Γ)) of a PST model set Λ(Γ) as the set of all Delone sets Λ0 locally isomorphic to Λ(Γ), which canalso be characterized as Ω(Λ(Γ)) = {t + Λ(Γ) : t ∈ Rd } [3].

2.2Discretized Primitive Tilings

A special family of PST model sets may be directly derived from primitive substitution tiling systems
defined as follows. Let P = {P1 , . . . , PN } be a set of tiles in Rd which are complements of quasilattices
Λ1 , . . . , ΛN in Rd such that Pi faces Pj for all 1 ≤ i, j ≤ N . A primitive substitution rule ω withscaling factor η > 1 on P is a rule which associates with Pi ∈ P a patch ω(Pi ) with support ηPi , whosetiles are translates of elements of P. Furthermore, there exists m ∈ N for which ω m (Pi ) contains atranslate of Pj for every 1 ≤ i, j ≤ N [2]. In this case, (P, ω) is called a primitive substitution
tiling system (PSTS). Additional assumptions listed in Remark 2.1 ensures the existence of a uniqueprimitive tiling space we denote as Ω(P,ω) containing every tiling obtainable from the PSTS.
Remark 2.1. Assume that all PSTSs (P, ω) have FLC and ω : Ω(P,ω) → Ω(P,ω) is one-to-one so thatΩ(P,ω) is compact, ω : Ω(P,ω) → Ω(P,ω) is a homeomorphism, and ω is locally invertible. This implies
that Ω(P,ω) is unique, compact, and contains no periodic tilings [2].
48

L. Silvestre et al.

Generating Frames via Discretized Substitution Tilings

TJCDCGGG2021

Every primitive tiling T based on a PSTS (P, ω) satisfying Remark 2.1 corresponds to an aperiodic
discrete set Ξ(T ) consisting of vertices of the tiles of T by considering its control points [1]. In this case,
we call Ξ(T ) a discretized primitive tiling, and define
o
n
(P,ω)
ΩΞ
= Ξ(T ) : T ∈ Ω(P,ω)(P,ω)

to be the discretized primitive tiling space associated with (P, ω). In fact, ΩΞis equal to the hull ofany discretized primitive tiling based on (P, ω).
The hole of any Λ ⊂ Rd is given by ρ(Λ) = sup inf d(λ, x). Given Λ ⊂ Rd with FLC and any fixedx∈Rd λ∈ΛSn
δ > 0, there exist finitely many disjoint translates {ti + Λ}ni=1 of Λ such that ρ( i=1 (ti + Λ)) <  [3]. In
this case, we call {ti }ni=1 a set of δ-translates of Λ.

2.3

Gabor Frames

The Gabor transform π of any h ∈ L2 (Rd ) with respect to x = (x1 , x2 ) ∈ Rd × Rd ∼= R2d is defined by
π(x)h(w) = e2πix2 ·w h(w − x1 ) for any w ∈ R.

For any function f with ||f ||L2 = 1 in the modulation space M1 (Rd ) = f ∈ L2 (Rd ) : ||Vf f ||L1 < ∞Rwhere Vf h(x) := Rd h(w)f (w − x1 )e−2πix2 ·w dw = hh, π(x)f i, a δ > 0 can always be chosen so thatαδ (f ) = sup x,y∈R2d ||π(x)f − π(y)f ||M1 = sup x,y∈R2d ||Vf (π(x)f − π(y)f )||L1 < 1.
|x−y|≤δ|x−y|≤δnWith these, the Gabor frame for L2 (Rd ) generated by {fi }i=1 ⊂ L2 (Rd ) and Λ ⊆ R2d is the collectionnG ({fi }i=1 , Λ) = {π(λ)fi : 1 ≤ i ≤ n, λ ∈ Λ} .

3Main Results

The following are the main results of the study characterizing model sets and discretized primitive tilings
generating Gabor frames.
N

Theorem 3.1. Let δ > 0. Suppose Λ(Γ) ⊂ R2d is a PST model set in R2d with FLC, and {ti = (ti1 , ti2 )}i=1 ne = S (ti + Λ(Γ)) satisfies ρ Λ
e < δ. Thenfor ti , ti ∈ Rd be a set of δ-translates of Λ(Γ) so that Λ
1

2

i=1
0

e and G(π(t1 )f, . . . π(tn )f, Λ0 )every Λ in the hull Ω(Λ(Ω)) of Λ(Γ) generates the Gabor frames G(f, Λ)2d1dfor L (R ), for any f ∈ M (R ) with αδ (f ) < 1.
Theorem 3.2. Let δ > 0 and (P, ω) be a primitive substitution tiling system in R2d satisfying Remark(P,ω)
2.1 with maxP ∈P {maxp∈PΛ d (cP , p)} = δ. Then every Λ ∈ ΩΞ
generates a Gabor frame G (f, Λ) withany window f ∈ M1 (Rd ) satisfying αδ0 (f ) < 1 for some δ 0 ≥ δ.
Theorem 3.3. Let δ > 0. Suppose f ∈ M1 (Rd ), and δ > 0 with which αδ (f ) < 1. If Λ ⊂ R2d is aδ
discretized primitive tiling with ρ(Λ) = ε, then G(f, Λ) is a Gabor frame if and only if ε ≤ |Λ∩B (0)|1/2d ·
εAcknowledgments. L. Silvestre acknowledges the support of the Ateneo de Manila University andthe Department of Science and Technology. J. Nable was supported by the SOSE ARISE Grant and an
RCW Grant, Ateneo De Manila University.

References[1] J. Lee, S. Akiyama, and Y. Nagai, Cut-and-project schemes for Pisot family substitution tilings, Symmetry.
10 (2018), 511.
[2] C. Starling, Actions of finite groups on substitution tilings and their associated C∗ -algebras, PhD Dissertation,
Université d’Ottawa, 2012.
[3] M. Kreisel, Gabor frames for quasicrystals, K-theory, and twisted gap labeling, J. Funct. Anal. 270 (2016),
1001–1030.

49

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Tiling of the Sphere by Congruent Polygons
Yohji Akama1 , Hoi Ping Luk2 , Erxiao Wang3 , and Min Yan4
1

Division of Mathematics, Faculty of Science, Tohoku University, Sendai 980-8579, Japan
Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China
3
College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
yoji.akama.e8@tohoku.ac.jp (Y. Akama), hoi@connect.ust.hk (H. Luk), wang.eric@zjnu.edu.cn (E.
Wang), mamyan@ust.hk (M. Yan)
2,4

Abstract
In an edge-to-edge tilings of the sphere by congruent polygons, the polygon is triangle, quadrilateral, or pentagon. The classification of triangular tilings was completed Ueno and Agaoka in 2002.
We almost completed the classification of pentagonal tilings, and have started the classification ofquadrilateral tiling. We survey these results, with emphasis on pentagonal tilings.
Keywords: spherical tiling, platonic solid, classification.
2010 MSC: Primary 05B45; Secondary 52B10.

1

Triangular Tilings

The classification of edge-to-edge tilings of the sphere by congruent triangles was started by Sommerville[7], and completed by Ueno and Agaoka [8]. The classification can be divided into two categories:• Construction from Platonic solids, and their modifications.
• Construction of the earth map kind, and their modifications.
The tilings in the first category have fixed number of tiles, and the tilings in the second category allow
arbitrary number of “timezones”.

2

Pentagonal Tilings

The pentagonal tilings has the following possible edge length combinations: a2 b2 c, a3 bc, a3 b2 , a4 b, a5 .
Here is the classification:
1. a2 b2 c [9, 11]: Three families of pentagonal subdivisions of the Platonic solids. Each family has two
free parameters.
2. a3 bc [9]: Two double pentagonal subdivisions of the Platonic solids. Each has unique pentagon.
3. a3 b2 [10]: (1) Five families of pentagonal subdivisions of the Platonic solids, each having one
free parameter; (2) For three special cases of (1), each with unique pentagon, we have ten flip
modifications of the pentagonal subdivision tilings.
4. a4 b [4, 5]: If the number of distinct angles is not four, then the tilings are (1) Three families ofpentagonal subdivisions of the Platonic solids, each with one free parameter; (2) Three families ofearth map tilings, and their flip modifications.
5. a5 [2]: (1) Three pentagonal subdivision tilings, each with unique pentagon; (2) Four earth map
tilings, each with unique pentagon; (3) Flip modification of one tilings from (2).
The only incomplete case is a4 b with four distinct angles.
The classification shows that the tilings can also be divided into two categories, similar to the triangular case.
50

Y. Akama et al.

3

Tiling of the Sphere by Congruent Polygons

TJCDCGGG2021

Quadrilateral Tilings

The classification can be divided into three kinds.
1. Kite, dart and rhombi [6]: These tilings can be reduced to triangular tilings. The classification
follows from [8].
2. Edge combination a2 bc: Completed by a team led by Erxiao Wang.
3. Edge combination a3 b: Only some preliminary results are obtained.
The quadrilateral tilings is the most difficult, and the difficulty is concentrated in the third kind.

References[1] Y. Akama and M. Yan, On deformed dodecahedron tiling, arXiv:1403.6907, 2014.
[2] Y. Akama, E. X. Wang, and M. Yan, Tilings of the sphere by congruent pentagons III: edge combination a5 ,to appear in Adv. in Math., arXiv:1805.07217, 2021.
[3] H. H. Gao, N. Shi, and M. Yan, Spherical tiling by 12 congruent pentagons, J. Combi. Theory Ser. A 120(2013), 744–776.
[4] H. P. Luk and M. Yan. Tilings of the sphere by congruent almost equilateral pentagons I: five distinct angles,
preprint, 2021.
[5] H. P. Luk and M. Yan. Tilings of the sphere by congruent almost equilateral pentagons II: three angles,
preprint, 2021.
[6] Y. Sakano and Y. Akama, Anisohedral spherical triangles and classification of spherical tilings by congruent
kites, darts and rhombi, Hiroshima Math. J. 45 (2015), 309–339.
[7] D. M. Y. Sommerville, Division of space by congruent triangles and tetrahedra, Proc. Royal Soc. Edinburgh
43 (1922-23), 85–116.
[8] Y. Ueno and Y. Agaoka, Classification of tilings of the 2-dimensional sphere by congruent triangles, Hiroshima Math. J. 32 (2002), 463–540.
[9] E. X. Wang and M. Yan, Tilings of the sphere by congruent pentagons I: edge combinations a2 b2 c and a3 bc,
to appear in Adv. in Math., arXiv:1804.03770, 2021.
[10] E. X. Wang and M. Yan, Tilings of the sphere by congruent pentagons II: edge combination a3 b2 , to appear
in Adv. in Math., arXiv:1903.02712, 2021.
[11] E. X. Wang and M. Yan. Moduli of pentagonal subdivision tiling, arXiv:1907.08776, 2019.
[12] M. Yan. Combinatorial tilings of the sphere by pentagons, Elec. J. of Combi. 20:#P1.54, 2013.
[13] M. Yan. Pentagonal subdivision, Elec. J. of Combi., 26:#P4.19, 2019.

51

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Renzuru Tilings with Asymmetric Quadrilaterals
Takashi Yoshino
Department of Mechanical Engineering, Toyo University, Kawagoe 350-8585, Japantyoshino@toyo.jp
Abstract
Renzuru, conjoined origami cranes, is one field of origamis. Most figures of Renzuru are basedon tilings with squares. However, the recent development of origami theory demonstrates thatthe origami cranes, constituents of Renzuru, can be folded from asymmetric quadrilaterals with
the inscribed circle. In this paper, we consider the extensions of the Renzuru by less symmetricalquadrilaterals than squares. We focus our consideration on the tilings with one shape in this study.
Therefore, we consider two cases: (a) the periodic tilings with congruent quadrilaterals and (b) thetilings with similar quadrilaterals. For the first case, we list ten types of tilings. Some tilings makepossible the 3-folded or 6-folded Renzurus without torsions. For the second case, the removal of thecentral part is necessary to fold Renzuru.
Keywords: Renzuru, origami cranes, tiling.
2010 MSC: Primary 52C20; Secondary 05B45, 51M04, 51M05, 01A27.

1Introduction

We consider the variations of tilings for Renzuru (連鶴), conjoined origami cranes. The traditional
Renzuru were summarized in the famous book titled “The Secret Heritage of How to Fold Thousands of
Cranes” (秘伝千羽鶴折形) published in 1797 [1]. The traditional origami crane is folded from a square
sheet so the traditional Renzuru is also folded from the sheet consisting of squares. An example ofRenzuru is shown in Fig. 1. The Renzuru was folded from a square sheet, and the sheet consisted of
3 × 3 smaller squares and was cut on the edges of small squares except for the narrow areas around thevertices.
On the other hand, recent researches revealed that the origami cranes could be folded from nonsquare paper [2, 3]. The result brings us the possibility of folding new types of Renzuru. In this paper,
we construct a list of quadrilateral tilings that enable us to fold Renzuru.

2

MethodWe defined the Renzuru tilings as satisfying both the two following conditions: one is that “origami
cranes could be folded ideally from all the constituted quadrilaterals”, and the other is that “no vertex

Figure 1: An example of Renzuru (conjoined origami cranes).

52

T. Yoshino

Renzuru Tilings

TJCDCGGG2021

was on the edge of other quadrilaterals”. The second condition was necessary to connect all origami
cranes. According to Justin [2] and Kawasaki [3], the necessary and suﬃcient conditions for a sheet
shape of origami cranes are that the quadrilateral having an inscribed circle. This condition is equivalent
to “the sums of opposite edges of the quadrilateral are equal”, so we used this condition as the first one
in the definition hereafter.
In this study, we focus our consideration on the tilings with one quadrilateral shape. It can beclassified into to cases; one is the tilings with congruent quadrilaterals and the other is the tilings withsimilar (but not congruent) quadrilaterals. In the first case, we used the complete list of the periodic
tilings with congruent convex quadrilaterals made by Grünbaum and Shephard [4]. According to thelist, the variation of the tilings is up to fifty-six. We examined all the items in the list if they satisfied
the required conditions. We referred to the tilings satisfying the Renzuru conditions as periodic Renzuru
tilings. In the second case, we considered the divisions based on logarithmic spirals. We constructed thedivision by using the points on the spirals as (ebtn cos tn , ebtn sin tn ) with tn = 2πn/k. For each value ofk(= 3, 4, 5, 6), we determined the value of b satisfying the condition for origami cranes.

3

Main Results

For the first case, the tilings with congruent quadrilaterals, we found ten types of periodic Renzuru
tilings. Figure 2 shows all the tilings. Eight tilings, from A to H, consist of the vertices of the degree offour. On the other hand, the rest tilings, I and J, include the vertices of degrees of three and six. They
are new types of conjoined origami cranes having 3-fold and 6-fold symmetries.
A

B

P4-56

F

C

P4-55

G

P4-46

P4-53

P4-51

P4-43

P4-48

J

I

H

P4-44

E

D

P4-42

P4-41

Figure 2: Periodic Renzuru tilings.

For the second case, we obtained four types of divisions as shown in Fig. 3. The figures correspondto the cases of diﬀerent values of k = 3, 4, 5, and 6 (from left to right). For all the cases, the removalsof the central part were necessary because these parts did not satisfy the condition for origami cranes.

Figure 3: Spiral Divisions.

References[1] M. Okamura, The World of Conjoined Cranes (in Japanese), Hon-no-Izumi Sha, Tokyo, 2006.
[2] J. Justin, Mathematical Remarks about Origami Bases, Symmetry: Culture and Science, 5 (1994), 153-165.
[3] T. Kawasaki, Roses, Origami and Math (in Japanese), Morikita Shuppan, Tokyo, 1998.
[4] B. Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman and Company, New York, 1987.

53

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Moving gold sand game
Akhmejanova Margarita1 , Bogdanov Ilya1 , and Chelnokov Grigory1,2
1

Moscow Institute of Physics and Technology, Laboratory of Combinatorial and Geometric Structure, Russia

2

National Research University Higher School of Economics, Moscow, Russia, E-mail:grishabenruven@yandex.ru(C. Grigory), mechmathrita@gmail.com (A. Margarita), ilya.i.bogdanov@gmail.com (B. Ilya)
Abstract
The paper deals with an algorithmic problem concerning combinatorial game theory. Here we
introduce and analyze a continuous generalization of Chip Game from [2]. The general Chip game wasintroduced by Aslam and Dhagat in [1] and models, in particular, on-line and list on-line coloringsof graphs and hypergraphs.
Keywords: on-line coloring, list on-line coloring, property B, Chip game, proper coloring, hypergraph coloring.
2010 MSC: Primary xxXxx; Secondary xxXxx, xxXxx, xxXxx.

1

Introduction

During the last two decades the algorithmic aspect of classic Erdős coloring problems [3] have received
new attention. For instance, property B was formulated in on-line version [1]. In this games one player
wishes to color all the elements so that the resulting coloring, for example, is proper and the other player
wishes to prevent this. Another motivation of this research is finding fast 2-coloring algorithms.

1.1

Chip game

There are actually a whole family of Chip games, first defined by Spenser [4]. Now we start with twovery similar versions of Chip game, which we called Chip Game1 (N, k) and Chip Game2 (N, k). Bothversions were created to model hypergraph and graph on-line coloring.
Chip Game1 (N, k): There is a board with positions −k, . . . , −1, 0, 1, . . . , k. In the starting arrangement only 0 position is occupied by N chips. Two players (Pusher and Remover) are playing. In eachround Pusher splits chips in each position into standing and running parts. Then Remover chooses one oftwo paths (positive or negative) and removes all running parts (except running part in 0 position) on thechosen path. Then Pusher moves all remaining running parts one step from 0 (towards −k if Remover
removed positive path and towards k, otherwise).Then a new turn begins. The game ends when either
there is a chip on the k or −k position (Pusher wins) or all chips are removed (Remover wins).
In 1993 Aslam and Dhagat in [1] proved that
Proposition 1.1. Pusher has a winning strategy in Chip Game1 (k · φ2k , k), where φ is a golden ratio
and Remover has a winning strategy in Chip Game1 (2k−1 , k).
Chip Game2 (N, k): There is a board with positions −k, . . . , −1, 0, 1, . . . , k. In the starting arrangement only k and −k positions are occupied, each of them by exactly N chips. Two players (Pusher
and Remover) are playing. In each round Pusher splits chips in each position into standing and runningparts. Then Remover chooses one of two paths (positive or negative) and removes all running parts on
the chosen path. Then Pusher moves all remaining running parts one step forward (towards 0). Then a
new turn begins. The game ends when either there is a chip on the 0 position (Pusher wins) or all chips
are removed (Remover wins).
In 2015 Duraj, Gutowski and Kozik in [2] proved that
Proposition 1.2. Pusher has a winning strategy in Chip Game2 (8 · 2k , k) and Remover has a winning
strategy in Chip Game2 (2k−1 , k).
Note that Proposition 1.2 gives much better bound (up to a multiplicative constant) for Pusher’s
strategy than does Proposition 1.1 with k · φ2k ≈ k(2.62)k .
54

A. Margarita et al.

2Moving gold sand game

TJCDCGGG2021

Main Results

Consider the following game:
Game1 (X, k): in each integer coordinate (we call it cell) i of the coordinate line OX lies a certain
amount xi of gold sand. Two players (Pusher and Remover) are playing. In each round Pusher splits gold
sand in each cell into two parts, standing and running (divides as he wants). Then Remover (knowing
how Pusher shared and knowing the amount of gold sand in each cell) chooses one of two directionsof the coordinate line OX (positive or negative) and removes all running parts on the chosen direction(road). Then Pusher moves all remaining running parts one step forward (towards 0). Moreover, allsand from cell 0 instantly becomes Pusher’s win.
The question: for a given arrangement X = (x−k , x−k+1 , . . . , xk ) find the supremum of Pusher’s
win.
Theorem 2.1. Consider the function g : [0; 1] 7→ R+ , defined by g(p) = x0 + x1 p + x2 p2 + . . . + pk xk +x−1 (1 − p) + x−2 (1 − p)2 + . . . + x−k (1 − p)k . Then the supremum of Pusher’s win is the minimum ofg(p) on [0, 1].
In case x = (x−k = 2k−1 , 0, . . . , 0, xk = 2k−1 ), i.e. in the continuous generalisation ofChip Game2 (2k−1 , k), our Theorem 2.1 says that supremum of Pusher’s win is equal to 1.
Theorem 2.2. Consider on-line version of Chip Game2 (N, k). Then supremum of Pusher’s win isagain minimum of g(p) on [0, 1].

3

Related work: chip game application

3.1

On-line property B

The on-line version of Erdős-Hajnal problem was firstly introduced by Aslam and Dhagat in [1]. For thesake of brevity, we describe it in slightly different form (just for r colors).
Game4 (k, N, r) is parametrized by three numbers: the cardinality of edges k, the number of colors r
and the number of edges N . Two players (Presenter and Colorer) are playing. In each round Presenter
presents one vertex and declares in which edges it is contained. He cannot add vertices to edges which
already contain k vertices. Then Colorer chooses any of r colors to the presented vertex. Then a newturn begins. The game ends when either there is a monochromatic edge (Presenter wins) or all N edges
contain k presented vertices each and no monochromatic edge (Colorer wins).
Let mol (k, r) denote the minimum N such that Presenter has a winning strategy in
Game4 (N, k, r). Clearly, mol (k, r) 6 m(k, r). From Duraj, Gutowski and Kozik’s work [2] follows thatmol (k, 2) 6 16 · 2k .
As an application of above results in [1] considered a Committee Meeting Problem: there are ncommittee members and s committees drawn from these members. At a conference there will be twogeneral meetings held in parallel. Is there a way to assign each member to one of these two meetings
so that each committee has at least one member at each meeting? Further, is it possible to make theseassignments on-line?
Acknowledgments. The authors acknowledge the financial support from the Ministry of Educationand Science of the Russian Federation in the framework of MegaGrant 075-15-2019-192 (Th. 2.1). The
first author is a Young Russian Mathematics award winner and would like to thank its sponsors andjury. The work of the first author also was also funded by RFBR, project number 20-31-90029 (Th. 2.2).

References[1] J. Aslam, A. Dhagat, “On-Line Algorithms for 2-Coloring Hypergraphs Via Chip Games.”, Theor. Comput.
Sci., 112:2, 1993, 355–369.
[2] L. Duraj, G. Gutowski, J. Kozik, “Chip games and paintability”, Electronic journal of combinatorics, 23:3,
2016, Article number P3.3.
[3] P. Erdős, A. Hajnal, “On a property of families of sets”, Acta Math. Acad. Sci. Hungar., 12:1, 1961, 87–123.
[4] J. Spencer. “Ten Lectures on the Probabilistic Method”, page 59. Society for Industrial and Applied Mathematics, 1987.

55

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Multi-Player Open-Hand Babanuki
Hironori Kiya1 and Hirotaka Ono2
1

Institute of Innovation for Future Society, Nagoya University, Nagoya 464-8601, Japan
Department of Mathematical Informatics, Nagoya University, Nagoya 464-8601, Japankiya.hironori@gmail.com (H. Kiya), ono@nagoya-u.jp (H. Ono)
2

Abstract
Babanuki is a card game played by two or more players. It is played in a hidden way, andthus it is rather a mind game than a mathematical game. In this paper, we propose an open-hand
Babanuki, which is a perfect information variant, in order to investigate its mathematical nature.
By this change, we have a trivial winning strategy for 2-player case, but it is not obvious for 3 or more
player’s cases. We find a necessary and sufficient condition of the existence of the winning strategy
for the 3-player case. In the 4-player case, we find that an endless-loop phenomenon, so-called“repetition draw”, occurs by all the players taking their own best strategies.
Keywords: multi-player combinatorial game, perfect information, winning strategy, repetition draw.
2010 MSC: Primary 91A46; Secondary , 91A06.

1Introduction

Babanuki is a popular game with playing cards in Japan. The rule is quite simple. In the setup phase,
all the cards including one joker are (evenly) distributed to the players. If a player has two cards with
an identical number in their hand, they discard them. Define a cyclic order of the players 0, 1, 2, . . .
(imagine a clock), and the game starts with player 1. Player 1 arbitrarily draws a card from the cards ofPlayer 0, which are put face down. A drawn card is added to the hand of Player 1, and if it forms a pair,they discard them as the setup phase. In the next turn, Player 2 similarly draws a card from the cardsof Player 1 and discards cards if possible; the game continues according to the cyclic (i.e., clock-wise)order. Note that the joker is never discarded since it is single. A player that gets to have no card leaves
the game, and the order of leaving is the ranking of the game. This is the basic rule of Babanuki .
Many similar games are played all over the world. For example, it is similar to card game “Old maid”,“Black Peter”, “Vieux Garçon”, “Le Pouilleux” or “Ekae” [1].
In this paper, we analyze Babanuki as a multi-player card-discarding game with perfect information. More precisely, we newly introduce “open-hand Babanuki ” based on the ordinary one. Here,“open-hand” means that players play Babanuki with cards put face up. Note that a p-player perfect
information game with p ≥ 3 does not always have an optimal strategy, though any two-player perfect
information game has an optimal strategy by the so-called Zermelo‘s theorem (see [2], for example). In
fact, the optimal strategy for the 2-player open-hand Babanuki is obvious; the player who does nothave the joker can always win just by drawing a card other than the joker. Thus, we focus on the caseswith 3 or more players.
Our results are summarized as follows: (1) We find a necessary and sufficient condition of the existenceof the winning strategy for the 3-player case. (2) In the 4-player case, we find that an endless-loop
phenomenon, so-called “repetition draw”, occurs by all the players taking their own best strategies. In
the remainder of the paper, we give a formal definition of open-hand Babanuki and a little more detailed
results in Sections 2 and 3, respectively.

2

Formal Model

We give the formal model of open-hand Babanuki . We use labels P0 , P1 , . . . , Pp−1 (p ≥ 2) for p players.
These labels are ordered in the cyclic manner, that is, (P0 , P1 , . . ., Pp−1 , P0 , . . . ). We emphasize that56

H. Kiya, et al.

Multi-Player Open-Hand Babanuki

TJCDCGGG2021

each Pi is not a name but a label for a player. As explained later, the label of a player may change
during a game, but we simply say “player Pi ” if no confusion arises, otherwise stated as “the player of
Pi ”. Note that p may decrease during a game. The set of card faces is [n] ∪ [n̄] ∪ {ι} for a positive integern, where [n] = {1, 2, . . . , n}, [n̄] = {1̄, 2̄, . . . , n̄} and ι is the joker. Notice that ι 6∈ [n] ∪ [n̄]. Here, card x¯ = x holds. For player Pi , let Hi denote the set of cards of Pi at the moment. If
and x̄ form a pair, and x̄
a player Pi has cards x and x̄ during a game, they can discard them; Hi is updated as Hi := Hi \ {x, x̄}.
A game proceeds as follows:(Setup phase) Distribute all the cards to players, which defines Hi for i = 0, . . . , p − 1. While some
Hi contains both x and x̄ for some x ∈ [n], update Hi := Hi \ {x, x̄}. Let i := 1 to represent turn,and k := 1 for ranking.
(Drawing phase) Player Pi draws an arbitrary card x from Hi−1 (Hp−1 for i = 0), that is, Hi−1 :=Hi−1 \ {x}. If Hi contains x̄, they are discarded (i.e., Hi := Hi \ {x̄}). Otherwise, just add x to
Hi (i.e., Hi := Hi ∪ {x}).
1. If Hi−1 = ∅, the player of Pi−1 leaves the game and is ranked as k-th. If i 6= p, reorder thelabels of players as Pi−1+j := Pi+j for j = 0, 1, . . . , p − i − 2. Update p := p − 1 and k := k + 1.
2. If Hi = ∅, the player of Pi is ranked as k-th leaves the game. If i < p − 1, reorder the labelsof players as Pi+j := Pi+j+1 for j = 0, 1, . . . , p − i − 2. Update p := p − 1 and k := k + 1.
3. Let i := i + 1 for i + 1 < p and i := 0 for i + 1 = p. If p = 1, the game ends. Otherwise, goback to the beginning of the drawing phase.
We assume that a smaller rank is better for every player. The result of a game depends on the initial
H0 , H1 , . . . , Hp−1 and the actions of all the players. The actions that player Pi can take are just to
select a card to draw from Hi+1 at Drawing phases. We define an optimal strategy of a player as a
strategy under which they can achieve the best rank among the achievable ranks. A player taking an
optimal strategy is called rational. In particular, we call a winning strategy a strategy by which a player
can get the first rank. We call a player having a winning strategy a definite winner. As mentioned in
Introduction, there is a game that has no definite winner for p-player open-hand Babanuki for p ≥ 3,
whereas 2-player one always has.

3

Results

We can construct an example that no player is a definite winner for 3-player open-hand Babanuki .
Showing the non-existence of a winning strategy is not obvious, since a case analysis requires to check an
exponential size of possible cases. For 3-player open-hand Babanuki, we find an explicit necessary andsufficient condition for the existence of a winning strategy, though we omit the detail due to the space
limitation. It yields the following theorem.
Theorem 3.1. In 3-player open-hand Babanuki, we can compute the definite winner O(n) time ifexists.
Since the configurations of p-player open-hand Babanuki include the configurations of (p − 1)-player
open-hand Babanuki, the situation becomes more complicated for more players. Indeed, there is agame where every player is never ranked due to an endless loop phenomenon for 4-player open-hand
Babanuki .
Lemma 3.2. Suppose that players P0 , . . . , P3 are rational and have hands such that H1 = H3 with
|H1 | = 2, H0 = H3 ∪ {ι} with |H3 | = 2, and the turn starts with P1 . Then, the optimal strategy of P1
is to draw ι of H0 .
By this lemma, P1 ’s action yields a new configuration that is essentially same as the original configuration by symmetry, which leads an endless loop.
Theorem 3.3. In 4-player open-hand Babanuki, there is a game that never ends by every rational
player drawing the joker in their turn.

References[1] pagat.com, Card game rules, https://www.pagat.com/, Accessed: 2021-07-15.
[2] Ulrich Schwalbe and Paul Walker, Zermelo and the early history of game theory, Games and economic behavior
34 (2001), no. 1, 123–137.

57

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Playing impartial games on a simplicial complex as extension ofemperor sum
Koki Suetsugu1
1

National Institute of Informatics, Chiyoda, Tokyo 101-8430, Japankoki.suetsugu@gmail.com (K. Suetsugu),
Abstract
In this study, we consider impartial games on a simplicial complex. On each vertex of given
simplicial complex, there is a position of an impartial game. A player, in their turn, choose a face ofthe simplicial complex and for each position on each vertex of the face, the player can make arbitrary
many moves. Moreover, the player can make at most one move for each position on each vertex notof the face. We show how we can characterize the P-positions of this game by using P-position
length. This result can be considered as an extension of the theory of emperor sum.
Keywords: Combinatorial Game Theory, impartial game, P-position length, Emperor Sum.
2010 MSC: Primary 91A46; Secondary 91A05, 05E45.

1Introduction

Combinatorial Game Theory studies two-player perfect information games with no chance moves. Impartial games are games in which both players have the same set of options in each position. In thisstudy, we consider that games are under normal play convention, that is, the player who makes the last
move is the winner. We also assume that games are loopfree, i.e., a position occurs at most once in a play.
Under these conventions, exactly one of the both players has a winning strategy at any given position.
We say a position in which the next (resp. previous) player has a winning strategy is an N -position(resp. P-position). An option of a position g is a position after one move from g, and a strict follower ofg is a position after arbitrary many moves from g.
Disjunctive sum of games is one of the most popular concept in Combinatorial Game Theory. For
any positions g and h, disjunctive sum of impartial games g + h is the game whose options are g 0 + h
and g + h0 , where g 0 and h0 ranges all options of g and h, respectively. That is, in disjunctive sum ofgames, a player chooses one component and make a move.
On the other hand, in this study, we consider the case that the player chooses multiple components
and makes multiple moves under certain conditions.

1.1

Early Results

Nim is one of the most famous impartial games. In this game, there are several heaps of stones. A playerchooses a heap and removes any positive number of stones from the heap. As we consider that games
are under normal play convention, the winner of nim is the player who removes the last stone. Bouton
proved that a position in nim is a P-position if and only if the bitwise XOR of the numbers of stones in
the heaps is zero [1].
Let V be a finite set of vertices. A simplicial complex ∆ on V is a subset of 2|V | such that for any
element v ∈ V, {v} ∈ ∆ and if F ∈ ∆ and G ⊆ F, then G ∈ ∆. An element of ∆ is called a face of ∆.
Ehrenborg and Steingrı́msson studied nim on simplicial complexes [3]. Let ∆ be a simplicial complex ona finite set V . Each vertex has some stones. In nim on ∆, a player, in their turn, chooses a non-empty
face F in ∆ and arbitrary removes any positive numbers of stones from all vertices in F . The original
nim, moore’s nim [4] and circular nim [2] can be considered as special cases of this game. Ehrenborg
and Steingrı́msson had not characterized P-positions of nim on ∆ with no restriction, but found good
constructions and characterized P-positions for certain cases.
We consider to expand Ehrenborg and Steingrı́msson’s study to general impartial games.
58

K. Suetsugu

Playing impartial games on a simplicial complex

TJCDCGGG2021

Definition 1.1. Let ∆ be a simplicial complex on a finite set V = (v1 , v2 , . . . , vn ) and G = (g1 , g2 , . . . , gn )be a collection of positions of impartial games.
Simplicial emperor sum of G on ∆, which is denoted by ∆(G), is a position such that on each vertex
vi , there is a position of an impartial game gi and a player, in their turn, chooses a face F in ∆ and makearbitrary many moves for positions on all vertices in F . In addition, for each position on the vertex not
in F , the player makes at most one move.
To characterize P-positions of this ruleset, we use P-position length.
Definition 1.2. For any position g, the P-position length of g is0,
Pl(g) =
max({Pl(g 0 ) : g 0 is a P-position and a strict follower of g}) + 1,

If g is a terminal position.
Othewise.

P-position length is used to characterize P-positions of emperor sum of games [5].
In emperor sum of games, a player selects one component and makes arbitrarily many moves. For
every other component, the player moves at most once. A position in emperor sum is a P-position if andonly if every component is an P-position and the bitwise XOR of P-position lengths of the components
is zero. Note that emperor sum is a special case of simplicial emperor sum on ∆ such that every face of∆ has single vertex.

2

Main Result

In this section, we discuss the way to determine which player has a winning strategy in simplicial emperor
sum. Let ∆ be a simplicial complex on a set V = (v1 , v2 , . . . , vn ) and P be the set of P-positions of nim
on ∆.
The following Lemmas are trivial from the definition of nim on a simplicial complex.
Lemma 2.1. Let A = (a1 , a2 , . . . , an ) ∈ P. For a position A0 = (a01 , a02 , . . . , a0n ), if there is a face F suchthat for any vi ∈ F, a0i < ai and for any vi 6∈ F, a0i = ai , then A0 is an N -position.
Lemma 2.2. For any B = (b1 , b2 , . . . , bn ) 6∈ P, there is a position B 0 = (b01 , b02 , . . . , b0n ) ∈ P such that aface F satisfies for any vi ∈ F, b0i < bi and for any vi 6∈ F, b0i = bi .
Theorem 2.3. For a collection of positions of impartial games G = (g1 , g2 , . . . , gn ), ∆(G) is a P-position
if and only if (Pl(g1 ), Pl(g2 ), . . . , Pl(gn )) ∈ P and for any i, gi is a P-position.
Sketch of the proof. Let X and Y be collections of positions of impartial games such that X = {(g1 , g2 , . . . , gn ) :(Pl(g1 ), Pl(g2 ), . . . , Pl(gn )) ∈ P and for any i, gi is a P-position.} and Y = {(g1 , g2 , . . . , gn ) : (g1 , g2 , . . . , gn ) 6∈
X}.
It is enough for proving this theorem to show following two claims.
Claim 1. For any G ∈ X, every G0 such that ∆(G0 ) is an option of ∆(G) satisfies G0 ∈ Y.
Claim 2. For any H ∈ Y, there is a collection of positions of impartial games H 0 such that H 0 ∈ X and∆(H 0 ) is an option of ∆(H).
Claim 1 and Claim 2 can be proved by using Lemma 2.1 and 2.2, respectively.
As emperor sum is a special case of simplicial emperor sum, this result can be considered as a
generalization of the theory of emperor sum of games.

References[1] C. L. Bouton, Nim, a game with a complete mathematical theory, Ann. of Math., 3 (1901), 35–39.
[2] M. Dufuor and S. Heubach, Circular Nim Games, Electron. J. Combin. 20(2)(2013), #P22.
[3] R. Ehrenborg and E. Steingrı́msson, Playing Nim on a simplicial complex, Electron. J. Combin., 3(1)(1996),#R9[4] E. H. Moore, A generalization of the game called Nim, Ann. of Math., Ser. 2 11 (1910), 93–94.
[5] K. Suetsugu, Emperor nim and emperor sum: a new sum of impartial games, Int. J. Game Theory,https://doi.org/10.1007/s00182-021-00782-0(2021)

59

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Losing Positions of Splythoff and 2-Splythoff encoded in the
Tribonacci word
Gerard Francis M. Ortega1
1 Department of Mathematics, Ateneo de Manila University, Katipunan Ave, Quezon City, 1108 Metro Manila, Philippines

gfmortega@hotmail.com

Abstract
We analyze 2-Splythoff, a generalization of Fokkink and Rust’s game of Splythoff, and show how
its P-positions are encoded in the Tribonacci word. Splythoff is itself a modification of Wythoff,allowing an extra move called a split. The table of letter positions of the Tribonacci word yields a
mex rule for 2-Splythoff that generalizes the result of Fokkink and Rust. We then show that thisgeneralized mex rule can be used to generate a new family of infinite words on a finite alphabet.
Keywords: combinatorial games, tribonacci word, integer sequence.
2010 MSC: 91A46; 37B10

1Introduction

Wythoff’s game is defined as follows. Two piles of counters are placed on the table. Two players
alternately either take an arbitrary number of counters from a single pile or an equal arbitrary number
from both piles; these moves are called singles or doubles, respectively. The player who takes the lastcounter, or counters, wins. Alternatively, the first player who cannot make a move loses.
To transform it into Splythoff, we allow for the option of a split. If a player performs a double whichleaves one of the piles empty, they may redistribute the remaining tokens to split that pile into two. For
example, from position (4, 7), it is possible to reach the position (1, 2); take 4 counters from each pile,then split the remaining counters into piles of size 1 and 2 [1]. To generalize to k-Splythoff, we loosen
the restrictions on a double move. A player may take x counters from one pile and y counters from theother if and only if |x − y| < k. The conditions for a split do not change.
Since k-Splythoff is finite and deterministic, the winner or loser from a given starting position should
be a foregone conclusion, assuming both players are behaving optimally, i.e. if it is possible for a player
to win, they always win. An N -position or winning position is one from which the current turn player
wins. A P -position or losing position is one from which the previous turn player wins.
The terminal position of (0, 0) is defined to be a P -position by the game rules. From there, the N
and P -positions can be recursively defined. A position is an N -position if there exists a move that leads
to a P -position. A position is a P -position if all moves from it lead to N -positions.

2

Preliminaries

We can enumerate the P -positions of Splythoff as follows. Let A and B be natural number sequencessuch that (ai , bi ) are the P -positions of Splythoff, where A < B (i.e., the smaller pile goes in A)∗ . We will
similarly enumerate the P -positions of 2-Splythoff using the sequences Â and B̂. We can then similarly
enumerate the P -positions of 2-Splythoff using the sequences Â and B̂.
Fokkink and Rust showed that the P -positions of Splythoff are encoded in the Tribonacci word.
Consider words over the finite alphabet {0, 1, 2}. The Tribonacci substitution τ sends 0 7→ 01, 1 7→ 02,
and 2 7→ 0. Repeated iterations of τ on 0 produces the sequence 0 7→ 01 7→ 0102 7→ 0102010 7→ . . . ,which then yields the infinite Tribonacci word t = 0102010010201 . . . in the limit.
∗ Following the convention of Fokkink and Rust, we denote infinite sequences by uppercase letters such as A, and denotetheir entries by a1 , a2 , a3 , . . . , using the corresponding lowercase letter

60

Gerard Francis M. Ortega

2-Splythoff and the Tribonacci word

A
B

1
1
2

2
3
5

3
4
8

4
6
11

5
7
13

6
9
16

7
10
19

TJCDCGGG2021

...
...

Figure 1: The first 7 terms of A and B.

Â
B̂

1
1
3

2
2
8

3
4
12

4
5
17

5
6
20

6
7
25

7
9
29

...
...

Figure 2: The first 7 terms of Â and B̂.
From this, we let X 0 , X 1 and X 2 be the natural number sequences that enumerate the (1-indexed)
positions which contain 0, 1, and 2, respectively, i.e. i ∈ X j if ti = j. This defines the positions tableof the Tribonacci word. Define the sequences ∆0 = X 1 − X 0 and ∆1 = X 2 − X 1 which comprisethe difference table of the Tribonacci word. Finally, let d∆ = ∆1 − ∆0 and s∆ = ∆0 + ∆1 be thedouble-difference and sum-difference sequences of the Tribonacci word.

X0
X1
X2
∆0
∆1
d∆
s∆

1
1
2
4
1
2
1
3

2
3
6
11
3
5
2
8

3
5
9
17
4
8
4
12

4
7
13
24
6
11
5
17

5
8
15
28
7
13
6
20

6
10
19
35
9
16
7
25

7
12
22
41
10
19
9
29

8
14
26
48
...
...
...
...

...
...
...

Figure 3: The first 8 terms of the positions table of the Tribonacci word. The difference table is also
shown, along with the double-difference and sum-difference sequences.
Fokkink and Rust showed the following connection between the Tribonacci word and Splythoff.
Theorem 2.1 (Fokkink and Rust). The ith P-position in Splythoff, (ai , bi ), is given by ai = (δ 0 )i andbi = (δ 1 )i , the rows of the difference table of the Tribonacci word [1].

3Main Results

Fokkink and Rust leave as an open question whether such a relationship exists for the general k-Splythoff.
Our main result is the following theorem for 2-Splythoff.
Theorem 3.1. The ith P-position in 2-Splythoff (âi , b̂i ) is given by âi = dδi and b̂i = sδi , the doubledifference and sum-difference sequences of the Tribonacci word.
We prove this by establishing a mex recurrence relationship between the sequences Â and B̂, andthen using properties of the Tribonacci word to show that this recurrence relationship is also satisfied byd∆ and s∆. This mex rule in 2-Splythoff is a natural generalization of a similar mex rule that appears in
Splythoff. Unfortunately, this pattern does not continue for k-Splythoff with k > 2; numerical evidence
leads us to conjecture that no substitution rule exists whose fixed point encodes the winning positionsof k-Splythoff for k > 2. Nonetheless, the generalized mex rule can be used to generate a new family of
infinite words, whose properties we also study in and of themselves.
Acknowledgments. The author would like to thank Dr Eden Miro, his undergraduate thesis advisor,
for her support and patience in the completion of this paper.

References[1] Robbert Fokkink and Dan Rust, A modification of Wythoff ’s Nim, arXiv e-prints (2019), arXiv:1904.08339.

61

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Linear-Time Rectilinear Drawings of Triconnected Subcubic
Planar Graphs with Orthogonally Convex Faces
Md. Manzurul Hasan1,3 , Debajyoti Mondal2 , and Md. Saidur Rahman1
1Graph Drawing & Information Visualization Laboratory,
Department of Computer Science and Engineering (CSE),
Bangladesh University of Engineering and Technology (BUET), Bangladesh.
2
Department of Computer Science, University of Saskatchewan, Canada.
3
Department of Computer Science, American International University-Bangladesh (AIUB), Bangladesh.
mhasan.cse00@gmail.com (M. M. Hasan), dmondal@cs.usask.ca (D. Mondal),saidurrahman@cse.buet.ac.bd (S. Rahman)

Abstract
A graph is called planar if it admits a planar drawing on the plane, i.e., no two edges create acrossing except possibly at their common endpoint. In a rectilinear drawing Γ of a planar graph,each vertex is drawn as a point and each edge is drawn as either horizontal or vertical line segment.
A face in Γ is called orthogonally convex if every horizontal or vertical line segment connecting
two points within the face does not intersect any other face. We examine the decision problem thattakes a planar graph as an input and seeks for a rectilinear drawing where the faces are drawn asorthogonally convex polygons. A linear-time algorithm for this problem is known for biconnectedplanar graphs, but the algorithm relies on complex data structures and linear-time planarity testing,which are challenging to implement. In this paper, we give a necessary and sufficient condition for atriconnected subcubic planar graph to admit such a drawing, and design a linear-time algorithm tocheck the condition and compute a desired drawing, if it exists. We also show that if a triconnectedsubcubic planar graph G admits a rectilinear drawing, then it must also admit a rectilinear drawing
with orthogonally convex faces.
Keywords: Graph drawing, Rectilinear Drawing, Orthogonally convex face, Subcubic graph.
2020 MSC: Primary 68R10; Secondary 05C85, 52C30, 52C35.

1Introduction

Rectilinear drawing is a popular graph drawing model that finds applications in VLSI floorplanning [4].
A rich body of research has examined this drawing style in the literature [4]. Initial works focused onplane graphs, i.e., a planar embedding of the graph is given as an input where the output must respect
the given embedding. Rahman et al. gave a necessary and sufficient condition for a plane graph to havea rectilinear drawing and an algorithm for finding a drawing if it exists [5] and developed a linear-timealgorithm based on this condition. Later, Rahman et al. gave a linear-time algorithm for triconnectedsubcubic planar graph [3]. Here a subcubic planar graph is a planar graph where the degree of every
vertex is at most three, and a graph is triconnected if the removal of any pair of vertices does notdisconnect the graph. In addition to the rectilinear constraints, researchers also focused on restricting
the shapes of the faces [1, 2]. Chang and Yen gave a linear-time algorithm that can decide whether aplane graph admits a rectilinear drawing with orthogonally convex faces, and compute such a drawing
if it exists [2]. Hasan and Rahman presented an algorithmic outline to extend the idea for biconnectedplanar graphs, but the technique is quite involved as it is based on linear-time planarity testing [4].
In this paper we give a simpler linear-time algorithm for computing a rectilinear drawing of a triconnected subcubic planar graph with orthogonally convex faces (if it exists). Our algorithm is based ona simple necessary and sufficient condition. We also prove that if a triconnected subcubic planar graphG admits a rectilinear drawing, then it must also admit a rectilinear drawing with orthogonally convex
faces.
62

M. M. Hasan et al.

Orthogonally Convex Drawing

TJCDCGGG2021

Figure 1: (a)-(b) Illustration of Condition (iv) of Theorem 2.1: 2-vertices are drawn by white circles.
(c)-(e) Illustration of Theorem 2.3, where a rectilinear drawing can be used to find a face that can beused as an outerface to compute a rectilinear drawing with orthogonally convex faces.

2

Rectilinear Drawings with Orthogonally Convex Faces

Preliminaries: For the basic graph theoretic terminologies we refer the readers to [4]. Let Γ be an
arbitrary planar embedding of a planar graph. A k-vertex is a vertex of degree k. A subdivision of anedge (p, q) in Γ is an operation that removes the edge (p, q) and adds a path (p, v1 , v2 , . . . , q) between p
and q. We refer to the path (p, v1 , v2 , . . . , q) as a chain. An edge which is incident to exactly one vertex
of a cycle C and is located outside of C is called a leg of C. The vertex of C to which a leg is incident
is called a leg-vertex of C. A cycle C in Γ is called a k-legged cycle of Γ if C has exactly k legs andthere is no edge which joins two vertices on C and is located outside C. Symmetrically, we can define
hand, hand-vertex, and k-handed cycle where the hands lie interior to C. If a 3-legged/handed cycle C
contains one or more 2-vertices, then C is called good, or else C is called bad.
Main Results: The main results are outlined in Theorems 2.1–2.3, but the proofs are omitted.
Theorem 2.1. Let G be a triconnected subcubic planar graph, and let Γ be an arbitrary plane embeddingof G. Then G has a rectilinear drawing with orthogonally convex faces if and only if Γ has a face F
satisfying the following conditions (i)–(iv): (i) There are at least four 2-vertices on F , (ii) F is containedin the inner subgraph ΓI (C) for any bad 3-legged cycle C in Γ, (iii) F is contained in the outer subgraphΓO (C) for any bad 3-handed cycle C in Γ, and (iv) there are at least two chains on the facial cycle for F inΓ. If there are exactly two chains P1 and P2 on F , and any of them, say P1 , contains exactly one vertex,
then the other face F 0 that contains P2 , has at least one 2-vertex which is not on P2 (Figure 1(a)-(b)).
Traversing the contours of all faces in Γ, one can check in linear time whether G satisfies the conditions(i) − (iv) of Theorem 2.1. If G satisfies the conditions (i) − (iv) of Theorem 2.1 then G has a planar
embedding Γ0 which satisfies the conditions by Chang and Yen [2], and hence using the drawing algorithmof Chang and Yen [2], one can also find a rectilinear drawing with orthogonally convex faces of the planar
embedding Γ0 of G in linear time. Thus the following theorem holds.
Theorem 2.2. Let G be a triconnected subcubic planar graph. Then we determine in linear time whether
G has a rectilinear drawing with orthogonally convex faces or not, and find such a drawing, if it exists.
Given a rectilinear drawing of G with a face F satisfying the condition by [3], we show a face F 0 existsthat satisfies the conditions of Theorem 2.1 (Figure 1(c)-(e)). Hence we have the following theorem.
Theorem 2.3. Let G be a planar triconnected subcubic graph. Then G has a rectilinear drawing withorthogonally convex faces if and only if G has a rectilinear drawing.

References[1] Md. Jawaherul Alam, Stephen G. Kobourov, and Debajyoti Mondal, Orthogonal layout with optimal facecomplexity, Comput. Geom. 63 (2017), 40–52.
[2] Y. Chang and H. Yen, On orthogonally convex drawings of plane graphs, Computational Geometry: Theory
and Applications 62 (2017), 34–51.
[3] M. S. Rahman N. Egi and T. Nishizeki, No-bend orthogonal drawings of subdivisions of planar triconnectedcubic graphs, IEICE Trans. Inf. & Syst. E88—D, No. 1 (January, 2005).
[4] Md. Manzurul Hasan and Md. Saidur Rahman, No-bend orthogonal drawings and no-bend orthogonally convex
drawings of planar graphs (extended abstract), Proc. of the 25th International Conference on Computing andCombinatorics (COCOON), LNCS, vol. 11282, Springer, 2019, pp. 254–265.
[5] M. S. Rahman, T. Nishizeki, and M. Naznin, Orthogonal drawings of plane graphs without bends, J. Graph
Algorithms Appl. 7 (2006), no. 4, 335–362.

63

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

A complete combinatorial characterization of greedy-drawable trees
Reiya Nosaka, Hiroyuki Miyata, and Shin-ichi Nakano
Faculty of Informatics, Gunma University, Japant201d062@gunma-u.ac.jp (R. Nosaka), hmiyata@gunma-u.ac.jp (H. Miyata), nakano@cs.gunma-u.ac.jp (S. Nakano)

Abstract
A (Euclidean) greedy drawing of a graph is a drawing in which, for any two vertices s, t, there is a neighbor vertex of sthat is closer to t than s in the Euclidean distance. Graph classes that admit greedy drawings have been actively studied
and greedy-drawable trees were characterized in terms of non-linear inequalities by Nöllenburg and Prutkin [4]. Using thecharacterization, they gave a linear-time recognition algorithm of greedy-drawable trees of maximum degree ≤ 4. However,a combinatorial characterization of greedy-drawable trees of maximum degree 5 is not known. In this paper, we give a
combinatorial characterization of greedy-drawable trees of maximum degree 5, which leads to a complete combinatorial
characterization of greedy drawable trees.
Keywords: graph drawing, greedy routing, non-linear inequalities
2010 MSC: 68R10, 65G30

1Introduction

The greedy routing is a routing protocol where a message isalways forwarded to the neighbor closest to the destination.
In the greedy routing, a message delivery might fail if themessage is forwarded to a node with no neighbor closer tothe destination. A graph drawing where the greedy routing
is guaranteed to succeed is called a greedy drawing. That is, a(Euclidean) greedy drawing of a graph is a drawing in which,for any two vertices s, t, there is a neighbor vertex of s thatis closer to t than s in the Euclidean distance.
Graph classes that admit greedy drawings have been actively studied since Papadimitriou and Ratajczak presented
a conjecture that every 3-connected planar graph admits agreedy drawing. Leighton and Moitra [3] and Angelini etal. [2] independently proved the conjecture of Papadimitriou
and Ratajczak. Leighton and Moitra [3] also investigated a
condition that a binary tree does not admit a greedy drawing.
Later, Nöllenburg and Prutkin [4] characterized greedy drawable trees in terms of non-linear inequalities. Using the characterization, they gave a linear-time recognition algorithmof greedy-drawable trees of maximum degree 4 and an explicit description of greedy-drawable binary trees. However,a combinatorial characterization of greedy-drawable trees ofmaximum degree 5 was not known.
In this paper, we give a combinatorial characterization (anexplicit description) of greedy-drawable trees of maximum degree 5. Since a careful analysis on the recognition algorithm
by Nöllenburg and Prutkin [4] easily leads to a combinatorial
characterization of greedy drawable trees of maximum degree≤ 4 and the maximum degree of a greedy drawable tree isless than 6, the result immediately leads to a complete combinatorial characterization of greedy drawable trees.

2

Preliminaries

Let T be a tree. A drawing Γ of T is called a straight-line
drawing if every node is represented as a point in the planeand every edge as the line segment between its endpoints.

We hereafter assume that all drawings are plane straight-line
drawings, that is, straight-line drawings with no edge crossing. The drawing Γ is said to be greedy if for every pair ofvertices s, t(s ̸= t) there exists a neighbor vertex u of s withd(u, t) < d(s, t), where d is a distance function. In this paper,
we consider the case where d is the Euclidean distance. If T
has a greedy drawing, we say that T is greedy-drawable.
ube the subtree of T that contains u obtained
Let Tuv
by deleting an edge uv and let huuv be the open half-planebounded by axis(uv) that contains u, where axis(uv) is theperpendicular bisector of uv. Angelini et al. [1] showed thatuis cona drawing Γ is greedy if and only if every subtree Tuv
tained in huuv in Γ.
Let r be a vertex of T and let v0 , . . . , vd−1 be the neighbors of r. Then, we consider a rooted tree Ti = (Vi , Ei )viwith the root r deﬁned
Trv+ rvi (i = 0, . . . , d − 1). Wei
T byw
let polytope(Ti ) := {huw | uw ∈ Ei , uw ̸= rvi , dT (w, r) <dT (u, r)}, where dT is the graph distance and let Te be thesubtree induced by the vertices r, v0 , . . . , vd−1 . Assume that
we have a greedy drawing of each Ti and want to construct a
greedy drawing of T by glueing the drawings. Then, the task
is to construct a drawing in which Te is drawn greedily andeach Tj (j ̸= i) is contained in polytope(Ti ). If polytope(Ti )is unbounded, we extend the two unbounded edges to the intersection of them and deﬁne the open angle of Ti , denotedby ∠Ti , as the open cone bounded by the resulted rays. (Theoriginal deﬁnition in [4] is slightly diﬀerent; but this deﬁnition works for our purpose). The angle of the cone is denotedby |∠Ti |. Nöllenburg and Prutkin [4] proved a remarkablelemma, called shrinking lemma, that states that if T has agreedy drawing in which each Ti has an open angle, thenthere is a greedy drawing of T in which each Ti is drawn inﬁnitesimally small, keeping the size of each open angle. Basedon this lemma, they proved that T has a greedy drawing with|∠Ti | < φi (i = 0, . . . , d − 1) if and only if there is a permutation τ on {0, . . . , d − 1} such that the following inequality64

R. Nosaka et al.

Greedy-drawable trees

TJCDCGGG2021

system has a solution. For i = 0, . . . , d − 1,0 ≤ αi , βi , γi ≤ 180,βi < αi , γi < αi ,βi + γi+1 < φτ (i) (i mod d),Figure 1: type-Bn tree

Figure 2: type-Cn tree

Figure 3: type-Dk,l,n tree

Figure 4: type-Ek,l,n tree(2.1)

αi + βi + γi = 180, α0 + · · · + αd−1 = 360,
sin(β0 ) · · · · · sin(βd−1 ) = sin(γ0 ) · · · · · sin(γd−1 ).
This inequality system describes the condition that Te isdrawn greedily and each open angle ∠Ti contains the subtrees Tj (j ̸= i). We do not explain further details here;
see [4] for more details.

3

Rooted trees that can be drawn with open
anglesNöllenburg and Prutkin [4] introduced a procedure to compute the tight upper bound on the maximum possible open
angle of a rooted tree. By analyzing their results carefully,
we give an explicit description of the rooted trees that can be
drawn with an open angle.
To describe the result, we introduce some deﬁnitions. Let
P be a path. A tree T is called a degree-k path associated to
P if T is constructed by attaching at most k −1 leaves to eachterminal vertex of P and at most k − 2 leaves to each internalvertex of P . The number of degree-k vertices is called theweight of T . A leaf of T that is adjacent to one of the terminal vertices of P is called an end leaf of T . We denote by|∠T |∗ the tight upper bound on the maximum possible open
angles of T . If T can be drawn with an open angle φ − ϵ forany ϵ > 0, but not φ, we write |∠T |∗ = φ−Figure 5: type-Fk,l,n tree

4

Main result

In this section, we describe a combinatorial characterization(an explicit description) of greedy drawable trees of maximum
degree 5.
Let T be a tree of maximum degree 5. Since removing
degree-2 vertices does not change greedy-drawability [4], we
assume that T does not contain a degree-2 vertex. Also,
since T cannot contain two degree-5 vertices if T is greedydrawable, we assume that there is only one degree-5 vertex r.
Let v0 , . . . , v4 be the neighbors of r and consider the rootedvi
tree Ti := Trv+ rvi (i = 0, . . . , 4) as in Section 2. If T isidrawn greedily, each Ti must be drawn with an open angle [4].
Thus we can decide greedy drawability of T by checking feasibitliy of the inequality system (2.1) in Section 2, setting
Proposition 3.1. Let T be a rooted tree with root r and φ0 = |∠T0 |∗ , . . . , φ4 = |∠T4 |∗ . By checking feasibility of theassume that T contains no degree-2 vertex. Then, T has a inequality system for each possible combination (T0 , . . . , T4 )greedy drawing with an open angle if and only if T satisﬁes (each Ti must satisfy one of the conditions in Proposition 3.1)
carefully, we can prove the following theorem.
one of the following conditions.
(A) T is single edge and r is one of the end vertices. In this Theorem 4.1. Let T be a tree of maximum degree 5 with nocase, we have |∠T |∗ = 180◦ and call T the type-A tree. degree-2 vertex. Then, T is greedy-drawable if and only if T
is a subgraph of a tree obtained by attaching trees T0 , . . . , T4
(B) T is a degree-3 path of weight n ≥ 1 and r is an end leaf described in Table 1 to single vertex.
of T . In this case, we have |∠T |∗ = (90◦ + 60◦ × 21n )−
T
T
T
T
T
and call T a type-Bn tree. See Figure 1.
A
A
A
B
D
0

A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
B1
B1

(C) T is a degree-4 path of weight n ≥ 1 and r is an end
leaf of T . In this case, we have |∠T |∗ = (120◦ × 21n )−and call T a type-Cn tree. See Figure 2.
(D) T is obtained by attaching a degree-4 path of weightn ≥ 1 and two degree-3 paths of weight k, l ≥ 1 anda leaf to single vertex v. r is one of the end leaves of
the attached degree-4 path that is farthest from v (withrespect to the graph distance). In this case, we have1|∠T |∗ = (45◦ × 21k + 30◦ × 21l )− × 2n−1
and call T a
type-Dk,l,n tree. See Figure 3.

1

A
A
A
A
A
A
A
A
B1
B1
B1
B1
B1
B2
B2
B2
B3
B1
B1

2

A
B1
B1
B1
B1
B2
B2
Bn
B1
B1
B1
B2
Bn
B3
B3
B4
B3
B1
B2

3

4

1

k,l,n

Bn
B1
B2
B3
Bn
B2
Bn
Bn
B1
B2
Bn
B4
Bn
Bn
B4
B5
B3
Bn
B2

D1,3,1 , C2 , F2,2 , E1,1,1 , F1,l
Dk,l,nE1,l,1 , F2,l , D1,1,2 , D2,2,1 , D1,l,1D1,l,1 , C2 , F2,2 , E1,1,1 , F1,l
D1,3,1 , C2 , F2,2 , E1,1,1 , F1,l
F1,l , D1,1,1
F1,2 , C1
C1 , F1,1
F1,4 , D1,1,1
F1,2 , C1
C1 , F1,1
C1 , F1,1
Bn
Bn
Bn
B5
Bn
Bn
Bn

(E) T is obtained by attaching a degree-4 path of weight Table 1: Subtrees of maximal greedy-drawable trees of maxn ≥ 1 and two degree-3 paths of weight k, l ≥ 1 to imum degree 5
single vertex v. r is one of the end leaves of the at- References
tached degree-4 path that is farthest from v (with respect to the graph distance). In this case, we have [1] P. Angelini, G. Di Battista, and F. Frati. Succinct greedy
drawings do not always exist. Networks 59(3):267–274, 2012.
|∠T |∗ = (60◦ × 21k + 60◦ × 21l )− × 21n and call T a type[2]P.
Angelini, F. Frati, and L. Grilli. An algorithm to construct
Ek,l,n tree. See Figure 4.
(F) T is obtained by attaching three degree-3 paths to single vertex v. r is one of the end leaves of the attacheddegree-3 paths that is not adjacent to v. Denoting by
k and r the weights of the two attached degree-3 paths
that do not contain r, we call T a type-Fk,l tree. In thiscase, we have |∠T |∗ = 60◦ × ( 21k + 21l )− . See Figure 5.

greedy drawings of triangulations. J. Graph Algorithms Appl.,
14(1):19–51, 2010.
[3] T. Leighton and A. Moitra, Some results on greedy embeddings in metric spaces, Discrete Comput. Geom., 44 (3): 686–
705, 2010.
[4] M. Nöllenburg, R. Prutkin. Euclidean greedy drawings oftrees. Discrete Comput. Geom. 58(3), 543–579, 2017.

65

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Number of Go positions on a connected graphChengyang Qian1
1

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
mike98@sjtu.edu.cn
Abstract
Let L(B) be the number of legal positions of the game Go played on a graph board B. When10
|B| goesan interval of length 10−5.9×10 which contains the lower limit of√ to infinity, we determine√|V (B)||V (B)|
L(B) and we find that
L(B) has upper limit 3. We also raise a heuristic algorithm that
calculates the exact value of the lower limit. When B is a tree, we propose an algorithm to compute
L(B) in O(|V (B)|2 log |V (B)|) time.
Keywords: graph Go, tree Go, position number.
2010 MSC: Primary 91A46; Secondary 05A99, 05C30.

1 Introduction()
A graph G is a pair of two sets V (G) and E(G) such that E(G) ⊆ V (G). We call V (G) and E(G) the2vertex set and the edge
setofG
respectively.
For
every
S
⊆
V
(G),
G[S]
refersto the graph with vertex( )set S and edge set S2 ∩ E(G). If a graph B is connected, we name it as a board. A position f of aboard B is a mapping from V (B) to {0, 1, 2}. In a position f , we name f −1 (0), f −1 (1) and f −1 (2) as
the uncolored vertices, the black stones and the white stones, respectively. Two stones x, y ∈ V (B) aresolidly connected in a position f if there exists i ∈ {1, 2} such that x, y belong to the same connectedcomponent of B[f −1 (i)]. A stone x ∈ V (B) is free in a position f if it is solidly connected to a stone
adjacent to an uncolored vertex. A position of a board is legal if all stones on the board
are free. We√
define L(B) to be the number of legal positions on the board, and let L(B) := |V (B)| L(B). An m × ngird board is the Cartesian product of two paths with number of points m and n respectively. We use
L(m, n) for the number of legal positions of the m × n grid board.
Go is a game between two players played on a 19 × 19 grid board, where the two players alternativelytransform the current legal position on the board to another legal position according to some rule. The
number of legal positions plays an important role in judging the diﬀiculty of a game.
The mathematical analysis of Go game and its variants attracts the attention of the researchers ofalgorithmic game theory [2,3]. Flammenkamp was the first to find the simulation results with the Monte
Carlo method back in 1992. However, it was not until 2016 that Tromp finally found the exact value ofthe possible legal positions of the original game Go [4, 5]. Tromp and Farnebäck [5] also considered legal
positions on an n × m grid board and proved that√lim mn L(m, n) = 2.9757 · · · .
n,m→∞Meanwhile, Farr [1] proposed the concept of Go polynomial to study the possibility of a randomly placedboard being legal. As a special case of both rectangle grid boards and tree boards, Weninger andHayward [6] studied the positional linear Go where the rectangle grid board is of size 1 × n. They tried
to solve the game for small n and found some properties on the strategies of the players.
We consider arbitrary boards, and we are interested in the growth rate of the number of legal positionsof boards when the number of vertices of this board tends to infinity.
Theorem 1.1.

• It holds thatlim

max L(B) = 3.

n→∞ |V (B)|=n• Let l3 = 2.551209391768942483074187315647971017948 · · · . Then it holds that
lim infminn→∞ |V (B)|=n

L(B) ∈ (l3 , l3 + 1.6 × 10−59099333243 ).

66

Chengyang Qian

Number of Go positions

TJCDCGGG2021

2 Estimating the number of Go positions
A tree is a connected acyclic graph. A rooted tree is a pair (T, RT ) where T is a tree and RT is a specifiedvertex of the tree known as the root. A leaf is a vertex with no children. The height Ht T of a rooted
tree T is the length of the longest path to a leaf from the root. We use RT h for the set of all rooted
trees with height h.
Let T be a rooted tree with its root RT . We use L0 (T ) for the number of positions of T where theroot RT has an unfree black stone and all stones are free except those that are solidly connected to theroot RT . Let us define
LL (T ) := L(T ) + L0 (T ),LU (T ) := L(T ) + 2L0 (T ).
√√For h ∈ N, let lh = inf T ∈RT h |V (T )| LL (T ) and uh = inf T ∈RT h |V (T )| LU (T ).
We need to establish the next two technical results for the proof of our theorem.
Proposition 2.1. It holds for all h ∈ N that lh < lh+1 < uh+1 < uh . Moreover, it holds
lim infmin

N →∞ |V (T )|=n

L(T ) = lim uh = lim lh .
h→∞

h→∞

√
|V (T )|

Proposition 2.2. Let Xh := arg minT ∈RT h
LL (T ). Then Xh is nonempty for all h ∈ N. When1 ≤ h ≤ 3, Xh contains exactly one element Xh ; Moreover, |V (X1 )| = 2, |V (X2 )| = 58, |V (X3 )| =4807557017275, and√
Xh = arg min |V (T )| LU (T )T ∈RT hfor h = 2, 3.
Proof of Theorem 1.1. Let Kn be the complete graph on n vertices. We can check that L(Kn ) = 3n − 1.
Obviously, L(T ) < 3|V (T )| for all boards T . Henceforth, we obtain√
lim max L(T ) = lim n 3n − 1 = 3.
n→∞ |V (T )|=nn→∞From Proposition 2.1 it follows
l3 < lim inf L(T ) < u3 .
|V (T )|→∞

With Proposition 2.2 we can calculate l3 and u3 and find that u3 − l3 < 1.6 × 10−59099333243 , and hence
we see that
lim inf min L(T ) ∈ (l3 , l3 + 1.6 × 10−59099333243 ).
n→∞ |V (T )|=n3 Algorithms
This part is omitted due to space constraint.
Acknowledgments. This work is supported by NSFC 11971305.

References[1] G. E. Farr, The Go polynomials of a graph, Theoret. Comput. Sci. 306 (2003), no. 1-3, 1–18.
[2] Graham Farr, The probabilistic method meets Go, J. Korean Math. Soc. 54 (2017), no. 4, 1121–1148.
[3] Graham Farr and Johannes Schmidt, On the number of Go positions on lattice graphs, Inform. Process. Lett.
105 (2008), no. 4, 124–130.
[4] John Tromp, The number of legal Go positions, Computers and games, Lecture Notes in Comput. Sci., vol.
10068, Springer, Cham, 2016, pp. 183–190.
[5] John Tromp and Gunnar Farnebäck, Combinatorics of Go, International Conference on Computers andGames, Springer, 2006, pp. 84–99.
[6] Noah Weninger and Ryan Hayward, Exploring positional linear Go, Advances in Computer Games, Springer,
2017, pp. 101–114.

67

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

The maximum numbers of the rigid faces and edges incontinuous ﬂattening processes of a polyhedron
Kazuki Matsubara1 and Chie Nara2
1

Faculty of Education, Saitama University, Saitama, 338-0002, Japan
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Tokyo, 164-8525, Japan
additivestructure@gmail.com (K. Matsubara), cnara@jeans.ocn.ne.jp (C. Nara)
2

Abstract
There are many ways to continuously ﬂatten polyhedra. We focus on the maximum number ofthe rigid faces, as well as the rigid edges, in continuous ﬂattening processes for a given polyhedron.
Some upper bounds of the maximum numbers are given for general polyhedra, and then for a regularpolyhedron, we provide a continuous ﬂattening process in which the number of rigid faces (edges)
attains the maximum, or gives a lower bound expected as the maximum.
Keywords: polyhedron, continuous ﬂattening, rigid face, rigid edge.
2010 MSC: Primary 52B05; Secondary 52C25.

1Introduction

We use the terminology polyhedron for a polyhedral surface in R3 that is permitted to touch itself
but without self-intersect. Flat folding of a polyhedron refers to its folding by creases, without selfcrossing, into a multi-layered ﬂat folded state with a ﬁnite number of creases. Continuous ﬂatteningof a polyhedron is ﬂat folding by a continuous process. There are many ways to continuously ﬂatten
polyhedra in the literature.
In applications of ﬂattening techniques of polyhedra to other ﬁelds, it may be better to make therigid area on its surface as large as possible. The authors in [3, 4] have discussed the numbers of rigid
faces and rigid edges during the ﬂattening processes for some speciﬁed polyhedra.

Figure 1: A ﬂattening process of a regular tetrahedron with two rigid faces (grey faces) and ﬁve rigid
edges (bold edges).
In this talk, we focus on the maximum numbers of rigid faces and rigid edges in ﬂattening processes
for a given polyhedron P .

2Preliminaries

First, we deﬁne the two numbers Nf (P ) and Ne (P ).
68

K. Matsubara et al.

The maximum numbers of the rigid faces and edges

TJCDCGGG2021

Deﬁnition 2.1. Let P be a given polyhedron and S be the set of all continuous ﬂattening processes of
P . Further let nf (P, ϕ) and ne (P, ϕ) be the numbers of rigid faces and rigid edges, respectively, in theﬂattening process ϕ of P . Then for P , the maximum numbers of rigid faces and rigid edges are deﬁnedby
Nf (P ) = max{nf (P, ϕ) | ϕ ∈ S}
andNe (P ) = max{ne (P, ϕ) | ϕ ∈ S},
respectively.
Figure 1 shows an example of a continuous ﬂattening process of a regular tetrahedron P . The example
follows that Nf (P ) ≥ 2 and Ne (P ) ≥ 5. On the other hand, the bellows theorem in [1] gives some upperbounds. Furthermore, by considering some properties of the continuous ﬂattening processes for a given
polyhedron P , we can obtain the following upper bounds of the numbers Nf (P ) and Ne (P ).
Lemma 2.2. Let P be a polyhedron. Let f (P ) and e(P ) be the numbers of faces and edges of P ,respectively. Then we have the following.
(1) Nf (P ) ≤ f (P ) − 1 and Ne (P ) ≤ e(P ) hold.
(2) If all the faces of P are triangular, thenNf (P ) ≤ f (P ) − 2, Ne (P ) ≤ e(P ) − 1;
moreover, if P has v ′ (P ) vertices at each of which the total angle of adjacent faces is smaller than
2π and can not be divided in halves by any subset of the faces, then
⌈ ′
⌉
⌈ ′
⌉
v (P )v (P )Nf (P ) ≤ f (P ) − 2+ ε, Ne (P ) ≤ e(P ) −,4
4
where ε = 1 when v ′ (P ) is 1 mod 4, and 0 otherwise.

3

Main Results

It seems diﬃcult to determine the values of Nf (P ) and Ne (P ) for a polyhedron in general. By investigating various ﬂattening methods as well as applying results in the literature, the numbers of Nf (P )and Ne (P ) can be determined or restricted for regular polyhedra.
Theorem 3.1.
(1) For a regular tetrahedron P , Nf (P ) = 2 and Ne (P ) = 5.
(2) For a regular octahedron P , Nf (P ) = 4 and Ne (P ) = 10.
(3) For a cube P , Nf (P ) = 3, 11 ≤ Ne (P ) ≤ 12.
(4) For a regular dodecahedron P , 2 ≤ Nf (P ) ≤ 6, 25 ≤ Ne (P ) ≤ 30.
(5) For a regular icosahedron P , 5 ≤ Nf (P ) ≤ 14, 21 ≤ Ne (P ) ≤ 27.
We think it is most likely that Ne (P ) = 11 for a cube P and Ne (P ) = 25 for a regular dodecahedron
P , but we have not yet proved these.

References[1] R. Connelly, I, Sabitov and A. Walz, The bellows conjecture, Beitr. Algebra Geom. 38 (1997), 1–10.
[2] E. D. Demaine and J. O’Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge
University Press, 2007.
[3] K. Matsubara and C. Nara, Continuous ﬂattening of multi-layered pyramids with rigid radial edges, J. Inf.
Process. 28 (2020), 834–840.
[4] J. Itoh and C. Nara, Continuous ﬂattening of the 2-skeletons in regular simplexes and cross-polytopes, J.
Geom. 110 (2019), No. 47.

69

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Minimizing Visible Edges in Polyhedra
Csaba D. Tóth1 , Jorge Urrutia2 , and Giovanni Viglietta3
1

Department of Mathematics, California State University Northridge, Los Angeles, CA; Department ofComputer Science, Tufts University, Medford, MA, USA. Partially supported by NSF DMS-1800734.
2
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, México. Partially
supported by PAPIIT IN105221, Programa de Apoyo a la Investigación e Innovación Tecnológica UNAM.
3
School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, Japan.
csaba.toth@csun.edu (C. D. Tóth), urrutia@matem.unam.mx (J. Urrutia), johnny@jaist.ac.jp (G. Viglietta)
Abstract
We prove that, given a polyhedron P in R3 , every point in R3 that does not see any vertex of P
must see eight or more edges of P; this bound is tight.
Keywords: Edge Guards, Polyhedron, Visibility Graphs.
2010 MSC: Primary 52; Secondary 52B05, 52B10.

1Introduction

A pivotal result in the study of art gallery or illumination problems is that every simple polygon withn vertices can be illuminated with at most b n3 c point lights placed at its vertices. If a simple polygon isorthogonal (i.e., its edges are parallel to the x- or y-axis), b n4 c vertices suffice; see [3, 4].
Not much is known about art gallery problems in R3 . We say that a point p in a polyhedron P sees apoint q if the line segment pq is completely contained in P. It is known that the vertices of a polyhedron
P do not necessarily see all points in the interior of P (see Figure 2a for an example). Upper and lower
bounds on the number of edges needed to illuminate a polyhedron in R3 are given in [1, 2, 5].
In this paper we show the following:
Theorem 1.1. Let P be a polyhedron in R3 . Any point p ∈ R3 that sees no vertices of P sees at least
eight of its edges. Moreover, any point p ∈ R3 sees at least six edges of P. These bounds are tight.
In the brush polyhedron depicted in Figure 1a, any interior point close to the tip of a tetrahedral“spike” sees exactly six edges. It is not hard to prove that any point p ∈ R3 sees at least six edges ofany polyhedron. We remark that this also holds if p is in the exterior of the polyhedron. Indeed, for anedge to be visible to p, it is sufficient that p sees one of its endpoints.

p(a) A brush(b) Constructing a Spherical Occlusion Diagram

Figure 1: Visibility in R3
When P is an orthogonal polyhedron, i.e., when all its faces have normal vectors parallel to one of
the coordinate axes, any point in the interior of P sees at least twelve edges (while any exterior point
sees at least eight edges). In fact, take the three planes through an internal point p with normal vectors
parallel to the coordinate axes; each of these planes intersects at least four edges visible to p. The boundis tight; it is achieved, for example, in a cube or in the polyhedron in Figure 2a.
70Viglietta et al.

2

Minimizing Visible Edges in Polyhedra

TJCDCGGG2021

Spherical Occlusion Diagrams

Proving that, for an arbitrary polyhedron in R3 , any point that sees no vertices can always sees at least
eight edges is tricky. To prove our results we study Spherical Occlusion Diagrams (SOD), introduced
in [6]. SODs arise from the visibility region of a point p that does not see any vertex of a polyhedron
P. Specifically, a SOD is the geometric figure obtained by orthographically projecting all visible edge
sub-segments onto a small sphere centered at p; see Figure 1b. What is obtained is a spherical (hence
planar) arrangement of noncrossing arcs of great circle, and the contact graph of these arcs is a simple
planar directed graph where each vertex has outdegree 2. Figure 2a shows the SOD corresponding to
the point p at the center of the polyhedron on the left.
Hence, a SOD is a set of noncrossing arcs of great circle on a sphere satisfying the following properties,
which are called diagram axioms in [6]: (a) if two arcs intersect, one “feeds into” the other; (b) each arc
feeds into two arcs; (c) all arcs that feed into the same arc reach it from the same side. These axioms
easily imply some properties of SODs:
Theorem 2.1. The following statements hold for every SOD. (1) No two arcs in a SOD feed into eachother. (2) Each arc in a SOD is shorter than a great semicircle. (3) Every SOD is connected. (4) A
SOD with n arcs partitions the sphere into n + 2 spherically convex regions.

(a) The SOD corresponding to the point p at the center of thepolyhedron on the left; the colors of the swirls indicate thebipartition of the swirl graph(b) A SOD with eight arcs andfour swirls

Figure 2: Some Spherical Occlusion Diagrams
A crucial structure in a SOD is a swirl, defined as a cycle of arcs such that each arc feeds into the next
going clockwise or counterclockwise. In the SOD in Figure 2a, the clockwise swirls are colored green,
and the counterclockwise ones are red. The swirl graph of a SOD is the undirected multigraph on theset of swirls such that, for each arc shared by two swirls, there is an edge in the swirl graph.
Theorem 2.2. The swirl graph of a SOD is a simple planar bipartite graph with nonempty partite sets;
moreover, every SOD has at least four swirls.
As a consequence we have the following corollary, which in turn implies our main result, Theorem 1.1:
Corollary 2.3. Any SOD has at least eight arcs.
Figure 2b shows an example of a SOD with exactly eight arcs and four swirls.

References[1] N. Benbernou, E. D. Demaine, M. L. Demaine, A. Kurdia, J. O’Rourke, G. T. Toussaint, J. Urrutia, and
G. Viglietta. Edge-Guarding Orthogonal Polyhedra. In Proceedings of the 23rd Canadian Conference on
Computational Geometry, Toronto, ON, pp. 461–466, 2011.
[2] J. Cano, C. D. Tóth, and J. Urrutia. Edge Guards for Polyhedra in Three-Space. In Proceedings of the 24th
Canadian Conference on Computational Geometry, Charlottetown, P.E.I., pp. 155–159, 2012.
[3] J. O’Rourke. Art Gallery Theorems and Algorithms, Oxford University Press, New York, NY, 1987.
[4] J. Urrutia. Art Gallery and Illumination Problems. In J.-R. Sack and J. Urrutia (Eds.), Handbook of Computational Geometry, North-Holland, Amsterdam, pp. 973–1027, 2000.
[5] G. Viglietta. Optimally Guarding 2-Reflex Orthogonal Polyhedra by Reflex Edge Guards. Computational
Geometry: Theory and Applications 86, article 101589, 2020.
[6] G. Viglietta. A Theory of Spherical Diagrams. arXiv:2107.05895, 2021.

71

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Flat folding problem with parallel creases with mountain-valley
assignment on a convex polygonal piece of paper
Hiro Ito1 and Sae Neshiba1
1

The University of Electro-Communications, Tokyo, Japanitohiro@uec.ac.jp (H. Ito), n2131129@gl.cc.uec.ac.jp (S. Neshiba)

Abstract
Deciding ﬂat foldability of a rectangular piece of paper with a given mountain-valley pattern is
known to be NP-complete. One special case known to be solvable in linear time is when the creases
are parallel to each other and to one side of the rectangular piece of paper; this case reduces to apurely one-dimensional folding problem. Recently, this case was generalized to a parallelogram of
paper: if the piece of paper is a parallelogram and creases are parallel to each other and to one sideof the paper, then it can be solved in linear-time. In this talk, we try to extend the result to a moregeneral shape of papers. That is, this talk shows that under the restriction of that all creases areparallel to each other, if the shape of the paper is any convex polygon, the ﬂat-foldability problem
with mountain-valley assignment can be solved in linear-time.
Keywords: origami, crease, ﬂat folding, convex polygon, algorithm

1

Introduction

A classic problem in computational origami is ﬂat foldability: given a crease pattern drawn on a piece of
paper, can the paper be folded ﬂat (into the plane) so as to have creases (folds by 180◦ ) exactly as speciﬁedby the crease pattern? In the variant considered here, we are also given a mountain-valley assignment,that is, a speciﬁcation of whether each crease should be folded in one direction (+180◦ or mountain)or the other (−180◦ or valley); together, the crease pattern and mountain-valley assignment constitutea mountain-valley pattern. Both versions of the ﬂat foldability problem — given a mountain-valley
pattern or just a crease pattern — are known to be NP-complete, even when the crease pattern consistsof horizontal, vertical, and diagonal creases on a rectangle of paper [1]. However, in the special casewhere the paper is a rectangle (intuitively, a long narrow strip) and all creases are perpendicular to twoof the sides as shown in Fig. 1, the piece of paper eﬀectively becomes a one-dimensional segment and ﬂat
foldability becomes tractable. Indeed, every crease pattern is ﬂat foldable with the alternating mountainvalley assignment, and ﬂat foldability of a speciﬁc mountain-valley pattern can be decided in linear
time [2,4]. We call this problem one-dimensional ﬂat foldability with mountain-valley assignment (1DFFMV for short). Recently a more-general case of ﬂat foldability with parallel creases on parallelogram
paper was solved: The all creases are parallel to each other and to one side of the paper, as shown in
Fig. 2. It was shown that this problem can be solved in linear time by extending the algorithm for
1DFFMV [3]. In this talk, we present that the algorithm can be extended to general convex polygons ofpaper.

Figure 1: An instance of 1DFFMV [2]Figure 2: An instance of the problem solved
by [3]

72

H. Ito and S. Neshiba2

Flat folding problem with parallel creases with mountain-valley
assignment on a convex polygonal piece of paper

TJCDCGGG2021

Main Results

We show the following theorem.
Theorem 2.1. The ﬂat folding problem with mountain-valley assignment in which the shape of thepiece of paper is a general convex polygon and all the creases are parallel to each other, can be solved inlinear-time.

Figure 3: An instance of the ﬂat folding problem with parallel creases with mountain-valley asignment
on a convex piece of paper

References[1] Hugo A. Akitaya, Kenneth C. Cheung, Erik D. Demaine, Takashi Horiyama, Thomas C. Hull, Jason S.
Ku, Tomohiro Tachi, and Ryuhei Uehara, Box pleating is head. In Revised Papers from the 18th Japan
Conference on Discrete and Computational Geometry and Graphs (JCDCGG 2015), Vol. 9943 of Lecture
Notes in Computer Science, pp. 167-179, Kyoto, Japan, September 2015.
[2] Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Martin L. Demaine, Joseph S. B. Mitchell, Saurabh
Sethia, and Steven S. Skiena. When can you fold a map? Computational Geometry: Theory and Applications,Vol. 29, No. 1, pp. 23-46, September 2004.
[3] Erik D. Demaine, Martin L. Demaine, Hiro Ito, Chie Nara, Izumi Shirahama, Tomohiro Tachi, Mizuho Tomura, Flat folding a strip with parallel or nonacute zig-zag creases with mountain-valley assignment, Special
Issue on Discrete and Computational Geometry, Graphs, and Games, Journal of Information Processing,Vol. 28, No. 12, pp. 825-833, 2020.
[4] Erik D. Demaine and Joseph O’Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra.
Cambridge University Press, July 2007.

73

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Max-norm analogs of Euclidean Ramsey theorems
Nóra Frankl1,2 , Andrey Kupavskii1,3 , and Arsenii Sagdeev1
1

Moscow Institute of Physics and Technology, Moscow, Russia
Carnegie Mellon University, Pittsburgh US
3
G-SCOP, Université Grenoble-Alpes, CNRS, France
nfrankl@andrew.cmu.edu (N. Frankl), kupavskii@ya.ru (A. Kupavskii), sagdeev.aa@phystech.edu (A.
Sagdeev)
2

Abstract
We find several analogs of classical Euclidean Ramsey results for spaces with the max-norm.
More concretely, we prove that, given a finite set S, the chromatic number of Rn with forbidden
S exponentially tends to infinity. Moreover, we find the base of this exponent for a large class offorbidden sets S and relate this to a problem of finding the optimal covering of Z by translated copies
of a given set. We also find an infinite set S such that the chromatic number of Rn with forbidden
S tends to infinity.
Keywords: Ramsey theory, chromatic number, coverings.
2021 MSC: Primary 05D10; Secondary 05D40.

1Introduction

There are several ways to generalize the well-known problem of Nelson about finding the chromatic
number χ(Rn ) of the n-dimensional Euclidean space. Euclidean Ramsey theory that lies at the junctionof geometry and Ramsey theory studies the following one. Given a subset S ⊂ Rd (with induced metric),
find the chromatic number χ(Rn ; S), i.e., the minimum number of colors needed to color all points of the
Euclidean space Rn with no monochromatic isometric copy S 0 ⊂ Rn of S. A set S is called Ramsey ifχ(Rn ; S) → ∞ as n → ∞.
A systematic study of such questions begins with the papers [1], [2] of Erdős, Graham, Montgomery,Rothschild, Spencer, and Straus. They showed that χ(Rn ; S) = 2 for all n ∈ N and all infinite S ⊂ Rn .
Moreover, they proved that each Ramsey set must also be spherical.
Relatively few sets are known to be Ramsey. Křı́ž [5] showed that each ‘fairly symmetric’ set isRamsey, e.g., the set of vertices of each regular polytope. Frankl and Rödl [4] proved that χ(Rn ; S) grows
exponentially in case S is a set of vertices of a box or a simplex. The best known lower and upper boundson χ(Rn ; S) for these S are relatively far from each other. For example, in the simplest two cases, when S
is a pair of points (clearly, in this case χ(Rn ; S) = χ(Rn )) or S is a set of vertices of an equilateral trianglennn∆ it is only known that (1.239... + o(1)) ≤ χ(Rn ) ≤ (3 + o(1)) and (1.01446... + o(1)) ≤ χ(Rn ; ∆) ≤nn
(2.732... + o(1)) , as n → ∞. The lower and upper bounds on χ(R ) are due to Raigorodskii [10] andLarman and Rogers [7] respectively. The lower and upper bounds on χ(Rn ; ∆) are due to Naslund [8](see also the paper [11] of the third author) and Prosanov [9] respectively.
In the present paper we give the analogs of all these statements for the n-dimensional spaces Rn∞
with the max-norm, defined for all x, y ∈ Rn as kx − yk∞ = max1≤i≤n {|xi − yi |} .

2

Main Results

In the upcoming paper [3] we prove that studying the values χ(Rn∞ ; S) for infinite S becomes more difficult
in comparison with the Euclidean case, where two colors are always enough for a proper coloring.
Theorem 2.1. For all n ∈ N and for all infinite S ⊂ Rn∞ , we have χ(Rn∞ ; S) ≤ n + 1. Moreover, for alln ∈ N, there is an infinite S ⊂ Rn∞ for which the last bound is tight.
74

N. Frankl et al.

Max-norm analogs of Euclidean Ramsey theorems

TJCDCGGG2021

Theorem 2.2. There is an infinite S ⊂ R such that χ(Rn∞ ; S) ≥ log3 n for all n ∈ N.
The last two authors [6] recently showed that all finite sets are Ramsey in the max-norm. Moreover,
for all d ∈ N and for all finite S ⊂ Rd∞ , there is a constant χ(S) > 1 such that χ(Rn∞ ; S) > (χ(S) + o(1))nas n → ∞. In the upcoming paper of all three authors [3] we find the optimal value χ(S) for several
families of forbidden sets S. The boxes are among the most interesting of them.
nTheorem 2.3. Let S ⊂ Rd∞ be a set of vertices of a box. Then χ(Rn∞ ; S) = (2 + o(1)) as n → ∞.
We also solve this problem for one-dimensional sets, but the precise statement requires some additional
notation. Let S ⊂ Z be a finite set. Then we call a subset A ⊂ Z to be an S-covering if S + A = Z,
where S + A = {s + a : s ∈ S, a ∈ A}. Similarly, a set A ⊂ Z is called (S, −S)-covering if both sets S + A
and −S + A coincide with Z. We define the values dc (Z; S) and dc (Z; S, −S) as the minimum possible
lower density of S-covering and (S, −S)-covering sets respectively. The following theorem relates thesenotions to the asymptotically tight bound on χ(Rn∞ ; S).
Theorem 2.4. Let S ⊂ Z be a finite set. Then χ(Rn∞ ; S) = (1 − dc (Z; S, −S) + o(1))−n as n → ∞.
Although the problem of finding the explicit expression for dc (Z; S) and dc (Z; S, −S) may be solved
for any fixed S by exhaustive search, it is probably very hard in general. In case |S| = 2 it is almost
obvious that dc (Z; S) = dc (Z; S, −S) = 21 . Schmidt and Tuller [12] stated a conjecture concerning thevalues of these functions for |S| = 3 (depending on S). We confirm their conjecture.
Theorem 2.5. Let λ1 , λ2 be two coprime positive integers and S = {0, λ1 , λ1 + λ2 }. Thendc (Z; S) = dc (Z; S, −S) = min 1d 3 (λ1 + 2λ2 )e d 13 (2λ1 + λ2 )e,.
λ1 + 2λ2
2λ1 + λ2Acknowledgments. The research was carried out at the expense of a grant from the Russian Science
Foundation (project 21-71-10092). Part of this research was done while the first author was a Visiting
Fellow at the London School of Economics with an LMS Early Career Fellowship. The third author is awinner of August Möbius Contest and Young Russian Mathematics Contest and would like to thank its
sponsors and jury.

References[1] P. Erdős, R.L. Graham, P. Montgomery, B.L. Rothschild, J. Spencer, E.G. Straus, Euclidean Ramsey theorems I, Journal of Combinatorial Theory, Series A, 14 (1973), N3, 341–363.
[2] P. Erdős, R.L. Graham, P. Montgomery, B.L. Rothschild, J. Spencer, E.G. Straus, Euclidean Ramsey theorems II, Colloq. Math. Soc. J. Bolyai, 10 (1973), Infinite and Finite Sets, Keszthely, Hungary and NorthHolland, Amsterdam, 520–557.
[3] N. Frankl, A. Kupavskii, A. Sagdeev, Max-norm Ramsey theorems, Unpublished manuscript.
[4] P. Frankl, V. Rödl, A partition property of simplices in Euclidean space, J. of Amer. Math. Soc., 3 (1990),
N1, 1–7.
[5] I. Křı́ž, Permutation groups in euclidean ramsey theory, Proceedings of the American Mathematical Society,112 (1991), N3, 899–907.
[6] A. Kupavskii, A. Sagdeev, All finite sets are Ramsey in the maximum norm, Forum of Mathematics, Sigma,
2021, to appear.
[7] D.G. Larman, C.A. Rogers, The realization of distances within sets in Euclidean space, Mathematika, 19(1972), 1–24.
[8] E. Naslund, Monochromatic Equilateral Triangles in the Unit Distance Graph, Bulletin of the London Mathematical Society, 52 (2020), N4, 687–692.
[9] R.I. Prosanov, Upper Bounds for the Chromatic Numbers of Euclidean Spaces with Forbidden Ramsey Sets,
Math. Notes, 103 (2018), N2, 243–250.
[10] A.M. Raigorodskii, On the Chromatic Number of a Space, Russian Math. Surveys, 55 (2000), N2, 351–352.
[11] A.A. Sagdeev, Exponentially Ramsey Sets Problems of Information Transmission, 54 (2018), N4, 372–396.
[12] W.M. Schmidt, D.M. Tuller, Covering and packing in Zn and Rn (I), Monatshefte für Mathematik, 153(2008), N3, 265–281.

75

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Celeste is PSPACE-hard
Lily Chung1 , Erik D. Demaine1
1

Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA
lkdc@mit.edu (L. Chung), edemaine@mit.edu (E. Demaine)

Abstract
We investigate the complexity of the platform video game Celeste. We prove that navigating
Celeste is PSPACE-hard in four different ways, corresponding to different subsets of the game mechanics.
Keywords: video games, hardness, PSPACE.
2010 MSC: 68Q17.

1Introduction

Celeste ∗ is a 2D platform video game released in 2018 by Extremely OK Games. It won the Best
Independent Game and Games for Impact awards at The Game Awards 2018 and sold over a million
copies [6]. In Celeste, the player controls a single character, Madeline, who must navigate various hazards
along her journey. We consider the following decision problem about Celeste:
Definition 1.1 (Celeste). Given a Celeste level, is it possible for Madeline to traverse from a designatedstart location to a designated end location?
We amend a previous result of Ahmed et al. [1] which attempted to show that Celeste is NPcomplete. We give four proofs that Celeste is instead PSPACE-hard, each using a different set ofgame mechanics. All of these proofs involve constructing a polynomial-time reduction to Celeste froma motion-planning problem through a planar network of doors [3]. We make use of both open-closetraverse doors, as introduced in [2, 4, 5] and shown not to need crossovers in [3], and self-closing doors,
as introduced in [3]. In all but one case we additionally show containment in PSPACE.

2

Main Results

Figure 1: Celeste entities mentioned in our results. From left to right: Madeline (the player, who canwalk, jump, climb, and dash), Spinner (spikes), Seeker (enemies), Jellyfish (reducing Madeline’s fall
speed), Pufferfish (can be bounced on), Move block (can slide in one direction until collision), Barrier(obstacle for enemies), Kevin block (can be moved in all directions, and reverses their path after collision).

Theorem 2.1. Celeste with spinners, seekers, barriers, and move blocks is PSPACE-complete.
∗ https://exok.com/games/celeste/. Celeste and its sprites are the properties of Extremely OK Games. Sprites areused here under Fair Use for the educational purpose of illustrating mathematical theorems.

76

L. Chung et al.

Celeste is PSPACE-hard

TJCDCGGG2021

Figure 2: Part of the proof of Theorem 2.1: an open-close-traverse door constructed with spinners, aseeker, barriers, and move blocks. Currently in the “closed” state.
See Figure 2 for the main gadget in the proof of Theorem 2.1.
Theorem 2.2. Celeste with spinners, jellyfish, and barriers is PSPACE-complete.
Theorem 2.3. Celeste with spinners and pufferfish is PSPACE-complete.
Theorem 2.4. Celeste with spinners and Kevin blocks is PSPACE-hard.
Other subsets of mechanics remain to be studied. In particular, it would be interesting to find subsetsthat are harder than PSPACE (e.g., undecidable), NP-complete, or (nontrivially) polynomial time.
Acknowledgments. This work was initiated during extended problem solving sessions with the participants of the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.892) taught by
Erik Demaine in Spring 2019. We thank the other participants for their insights and contributions. In
particular, we thank Dylan Hendrickson and Josh Brunner for helping simplify the proof of Theorem 2.4,
and we thank Michael Coulombe and Jayson Lynch for reading and suggesting revisions to drafts of thepaper.
Images of sprites and gadgets were composed and tested using the fan-made level editor Ahorn† . We
thank Cruor, Vexatos, and Ahorn’s other contributors for creating this excellent tool. We additionally
thank the Celeste speedrunning, modding, and Tool-Assisted Speedrunning community for extensively
researching Celeste’s mechanics.
Finally, we thank Extremely OK Games for producing Celeste, a difficult and wonderful experiencein many ways.

References[1] Zeeshan Ahmed, Alapan Chaudhuri, K. Grover, Hrishi Narayanan, Manasvini Vaidyula, and Shreyas Pradhan,Games and computational complexity, arXiv:2012.07678, 2020, https://arXiv.org/abs/2012.07678.
[2] Greg Aloupis, Erik D. Demaine, Alan Guo, and Giovanni Viglietta, Classic Nintendo games are (computationally) hard, Theoretical Computer Science 586 (2015), 135–160.
[3] Joshua Ani, Jeffrey Bosboom, Erik D. Demaine, Yevhenii Diomidov, Dylan Hendrickson, and Jayson Lynch,
Walking through doors is hard, even without staircases: Proving PSPACE-hardness via planar assemblies ofdoor gadgets, Proceedings of the 10th International Conference on Fun with Algorithms (FUN 2020) (La
Maddalena, Italy), September 28–30 2020, pp. 3:1–3:23.
[4] Giovanni Viglietta, Gaming is a hard job, but someone has to do it!, Theory of Computing Systems 54 (2014),
no. 4, 595–621.
[5], Lemmings is PSPACE-complete, Theoretical Computer Science 586 (2015), 120–134.

[6] Wikipedia, Celeste (video game), https://en.wikipedia.org/wiki/Celeste_(video_game).
† https://github.com/CelestialCartographers/Ahorn

77

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

NP-Completeness of Peg Solitaire on Graphs
Kazushi Ito1 , and Yasuhiko Takenaga1
1

Department of Computer and Network Engineering, Graduate School of Informatics and Engineering,
The University of Electro-Communications 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japantakenaga@cs.uec.ac.jp (S. Author)

Abstract
Peg solitaire is a single-player table game. The goal is to remove all but one peg from the
gameboard. Peg solitaire is generalized to be played on arbitrary graphs. A graph is called solvable
if there exists some vertex s such that the graph can be solved starting with s as the initial hole. We
prove that it is NP-complete to decide the solvability of a graph.
Keywords: peg solitaire, game, graph, NP-completeness.
2010 MSC: Primary 68Q17; Secondary 91A43, 05C57.

1

Introduction

Peg solitaire is a single-player table game which has been played for centuries. The goal of the game isto remove all but one peg from the gameboard. Peg solitaire usually starts from a board with only one
initial hole (a vertex without a peg). If there are three adjacent vertices x, y and z in a row or a column
and pegs are placed on x and y, and not on z, a peg on x can jump over the peg on y and be placed onz. The peg on y is removed. The problem to decide if the player can solve the given situation on peg
solitaire is known to be NP-complete [1].
Since Beeler and Hoilman generalized peg solitaire to be played on graphs in [2], solvability of peg
solitaire on several graphs has been studied (e.g. [3, 4]). A graph is called solvable if there exists somevertex s such that the graph can be solved starting with s as the initial hole. In this paper, we provethat it is NP-complete to decide the solvability of a graph.

2

NP-CompletenessFirst, we show the NP-completeness of the problem to decide the solvability when an initial hole is given
together with a graph. After that, we show the NP-completeness when the initial hole can be chosen
arbitrarily.
Theorem 2.1. Given a graph and an initial hole, it is NP-complete to decide if the graph is solvablewith respect to the initial hole.
Proof. NP-hardness is proved by reduction from the Hamiltonian circuit problem for planar directed
graphs with degree three [5]. Let G be the given directed graph and H be the graph obtained by the
reduction. For clarity, we use terms ‘node’ and ‘arc’ for G, and terms ‘vertex’ and ‘edge’ for H. We canassume that a node is either with indegree two and outegree one, or with indegree one and outdegree two.
A node gadget representing a node with indegree one is shown in Fig.1. A node gadget with indegree
two is the same as the graph except that the direction of the corresponding arcs are reversed. Nodegadgets are directly connected by an edge if an arc exists between two nodes. The initial hole is vertex
v9 of an arbitrary gadget representing a node with indegree one.
To solve H when G has a Hamiltonian circuit, visit the node gadgets in the order they appear in thecircuit (or in the reversed order). Pegs on the corresponding circuit in H can be removed with moves
such as u11 → u10 → u9 , u8 → u9 → u10 , u13 → u12 → u11 , u10 → u11 → u12 , · · · . During the process to78

K. Ito et al.

NP-Completeness of Peg Solitaire on Graphs

TJCDCGGG2021

o₅v₁

v₉ v₁₀

v₇

v₁₇

o₁
oʼ₁

oʼ₅

Figure 1: A node gadget with indegree one.

b5

b1

c5

c1

a0 a1

ai ai+1 ai+2

Figure 2: Part of a gadget.

remove pegs on the corresponding circuit in H, five pegs that are not on the circuit can also be removed
in each gadget.
Next, consider the case that G does not have a Hamiltonian circuit. Let Fig.2 be a part of the graphobtained by the reduction. Among the vertices shown in the figure, only vertices ai , ai+1 and ai+2 (i ≥ 2)
are holes. To move a peg to ai+1 , it is necessary to remove pegs from a0 , · · · ai . Thus, it is not possible
to move a peg to ai+2 from the left.
It can be checked that, during the process to remove pegs in a node gadget, at most five pegs that are
not on the path can be removed, that is, the pegs on the other gadgets cannot be removed. Therefore,
if a series of moves enters a node gadget from an incoming (resp. outgoing) arc, it must go out from an
outgoing (resp. incoming) arc. We can also see that it is not possible to fork the process of removing
pegs into two directions.
Theorem 2.2. Given a graph, it is NP-complete to decide if the graph is solvable.
Proof. The reduction is obtained with a small modification of the one in Theorem 2.1. We only add
eight vertices to v7 and v10 as a path, respectively, on a node gadget with indegree one. If G has a
Hamiltonian circuit, the reduced graph can be solved by choosing an appropriate vertex on the pathfrom v10 as the initial hole. If the initial hole is not on the path, at least one peg remains on the path.
Thus, to solve the graph, the initial hole is on one of the added paths and the last peg remains on theother added path.
Acknowledgments. This work was supported by JSPS KAKENHI Grant Number 18K11601.

References[1] R. Uehara and S. Iwata, Generalized Hi-Q is NP-complete, The Transactions of the IEICE, E73-E(2) (1990)
270-273.
[2] R. A. Beeler and D. P. Hoilman, Peg solitaire on graphs, Discrete Mathematics, 311 (2011) 2198-2202.
[3] R. A. Beeler and A. D. Gray, Freely solvable graphs in peg solitaire, ISRN Combinatorics 2013, Article ID
605279 (2013).
[4] J. Engbers and C. Stocker, Reversible peg solitaire on graphs, Discrete Mathmatics 338 (2015) 2014-2019.
[5] J. Plesńik, The NP-completeness of the Hamiltonian cycle problem in planar digraphs with degree boundtwo, Information Processing Letters, 8(4) (1979) 199-201.

79

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Prime-Graceful Number
Sayan Panma1 , and Penying Rochanakul1
1

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
panmayan@yahoo.com (S. Panma), Penying.Rochanakul@cmu.ac.th (P. Rochanakul)

Abstract
A graph G with n vertices and m edges, is said to be k-prime-graceful, if there is an injection
ψ : V (G) → {1, 2, ..., k min{m, n} + 1}, where gcd(ψ(u), ψ(v)) = 1 for all e = {u, v} ∈ E(G) andthe induced function ψ ∗ : E(G) → {1, 2, ..., m} defined as ψ ∗ (e) = |ψ(u) − ψ(v)| is injective. The
prime-graceful number of graph G is the minimum k such that G is k-prime-graceful.
In this research, we introduce k-prime-graceful labeling and the prime-graceful number, thenshow that prime-graceful numbers of star K1,n , bistar Bn,n , bistar Bn,p−2 , where p is an odd prime,(3)complete bipartite graph K2,n , triangular book graph Bn and some spiders are equal to 1.
Keywords: graph labeling, prime-graceful labeling, k-prime-graceful, prime-graceful number.
2010 MSC: Primary 05C78; Secondary 05C30, 05C90.

1 Introduction
Graph labeling is one of the active fields in graph theory. There are numerous applications of graphlabeling, including graph decomposition, coding for radar and missile guidance, X-ray crystallographic
analysis, designing communications networks addressing, determining the optimal circuit layout, etc.
More applications and details can be found in References [1–3].
A labeling of a graph is an assignment of labels to vertices and/or edges of the graph. The conceptof graph labeling was first introduced by Rosa [7] in 1967. Since then, hundreds of graph labelings havebeen studied. Gallian has made a thorough survey on those labelings and gather them in a dynamic
survey of graph labeling [4]. The term graceful labeling was first mentioned by Golomb [5] in 1972.
While the term prime labeling was introduced by Tout, Dabboucy, and Howalla [9] in 1982.
In this research, we introduce the k-prime-graceful labeling which is a combination of graceful labelingand prime labeling, then define the prime-graceful number.

2 Preliminaries
The notion of prime graceful graph was first studied by Selvarajan and Subramoniam [8], where we
renamed their work as 2-prime-graceful labeling. Later, Pavithra and Mary [6] extended the idea to
3-prime graceful labeling. Here are our generalize versions.
Definition 2.1. For a positive number k, a k-prime-graceful labeling of a graph G = (V, E) with nvertices and m edges is an injective function ψ : V (G) → {1, 2, . . . , k min{n, m} + 1} with the following
property: for any edge e = {u, v} ∈ E(G),1. the value gcd(ψ(u), ψ(v)) = 1,
2. the induced function ψ ∗ : E(G) → {1, 2, . . . , m}, defined as ψ ∗ (e) = |ψ(u) − ψ(v)|, is injective.
A graph is called k-prime-graceful if it has a k-prime-graceful labeling.
Note that, when k = 1, we omit the k and call them prime-graceful labeling and prime-graceful graph.
It is interesting to find the smallest number k of each graph. Here we give the notion of the primegraceful number.
80

S. Panma et al.

Prime-Graceful Number

TJCDCGGG2021

Definition 2.2. The prime-graceful number of G, denoted ξ(G), is the minimum k such that G is
k-prime-graceful.
Thus, we can conclude from prior works that the prime-graceful number of path Pn , cycle Cn , star
K1,n , friendship graph Fn , bistar Bn,n , C4 ∪ Pn , Km,2 and Km,2 ∪ Pn are at most 2. While the primegraceful number of fan graph Fn , wheel graph Wn , helm graph Hn , gear graph Gn , flower graph F ln ,sunflower graph SFn , closed helm graph CHn , and web graph W bn are at most 3.

3 Main Results
In this section, we show that star K1,n , bistar Bn,n , bistar Bn,p−2 , where p is an odd prime, complete(3)
bipartite graph K2,n , triangular book graph Bn and some spiders have prime-graceful number equal to1.
Theorem 3.1. Stars K1,n are prime-graceful.
Corollary 3.2. For any star K1,n , ξ(K1,n ) = 1.
Theorem 3.3. For n ∈ N, the bistar graph Bn,n is prime-graceful.
Theorem 3.4. For n ∈ N and a prime p > 2, the bistar graph Bn,p−2 is prime-graceful.
Corollary 3.5. For any bistar Kn,m , ξ(K1,n ) = 1 if m = n or m + 2 is an odd prime.
Theorem 3.6. For n ∈ N where n > 2, the spider of n legs where all of its legs have lengths two isprime-graceful if and only if 2n + 1 or 2n + 3 is prime.
Theorem 3.7. For n ∈ N, the complete bipartite graph K2,n is prime-graceful.
Corollary 3.8. For any complete bipartite graph K2,n , ξ(K2,n ) = 1.
(3)

Theorem 3.9. For n > 1, the triangular book graph Bn is prime-graceful if and only if n + 2 is an odd
prime or n + 1 is a power of two.
(3)

(3)

Corollary 3.10. For any triangular book graph Bn with n > 1, ξ(Bn ) = 1 if and only if n + 2 is an
odd prime or n + 1 is a power of two.
Acknowledgments. This work is supported by the Research Fund for DPST Graduate with First
Placement [Grant no. 038/2015], The Institute for the Promotion of Teaching Science and Technology(IPST), Thailand.
The authors are grateful to the referees for their careful reading of the manuscript and their usefulcomments.

References[1] B.D. Acharya, S. Arumugam, A. Rosa, Labelings of Discrete Structures and Applications, Narosa, 2007.
[2] G. S. Bloom, S. W. Golomb, Applications of numbered undirected graphs, in Proceedings of the IEEE 65 (4)(1977) pp. 562–570.
[3] G. S. Bloom, S. W. Golomb, Numbered complete graphs, unusual rulers, and assorted applications, Alavi Y.,
Lick D.R. (eds), Theory and Applications of Graphs. Lecture Notes in Mathematics 642, Springer, Berlin,
Heidelberg, 1978.
[4] J.
Gallian,A
dynamic
surveyofgraphlabeling,
Electron.
http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6/pdf.

J.

Combin.

#DS6,[5] S. W. Golomb, How to number a graph, in Graph Theory and Computing, R. C. Read, ed., Academic Press,
New York (1972) 23–37.
[6] P. Pavithra, U. Mary, 3-Prime Graceful Labeling, Journal of Information and Computational Science, 10(1),(2020) 1381–1387.
[7] A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome,
July 1966), Gordon and Breach, N. Y. and Dunod Paris (1967) 349–355.
[8] T. M. Selvarajan, R. Subramoniam, Prime Graceful Labeling, International Journal of Engineering & Technology, 7(4.36) (2018) 750–752.
[9] A. Tout, A. N. Dabboucy, K. Howalla, Prime labeling of graphs, Nat. Acad. Sci. Letters, 11 (1982) 365–368.

81

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Rainbow Connection Number of Dutch Windmill Graph
Aroonwan Suebsriwichai1 and Thanasak Mouktonglang2
1

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University,
Pathumthani 12121, Thailand
2
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
aroonwan@mathstat.sci.tu.ac.th, thanasak.m@cmu.ac.th

Abstract
The rainbow connection number of G, denoted rc(G), is defined as the minimum number ofcoloring edges in G where every two vertices u and v in V (G) is connected by at least one path forwhich no edge are colored the same. In this paper, we determine the rainbow connection number(m)(m)
of the dutch windmill graph Dn . We also describe a method of coloring the edges in Dn as arainbow-connected graph.
Keywords: Rainbow connection number, dutch windmill graph, coloring.
2010 MSC: 05C15, 05C38, 05C40.

1Introduction

Let G be a simple, finite, undirected and connected graph with a vertex set V (G) = {v1 , v2 , ..., vn } and anedge set E(G) = {e1 , e2 , ..., em }. The rainbow connection number concept was introduced by Chartrand
et al. [1] in 2008. A coloring of the edge of G, denoted c, is a function from its edge set to the set ofthe positive integer k, that is c : E(G) → {1, 2, ..., k} such that the adjacent edges can be colored thesame. We call a u − v path P in G as a rainbow path, if all edges in path P receive distinct colors. The
graph G is called rainbow-connected, if there exists a rainbow u − v path for every two vertices u and v
of G. The rainbow connection number of G, denoted rc(G), is defined as the minimum number of colorsof rainbow-connected graph.
In 2004, Yixiao Liu and Zhiping Wang [2] announced the rainbow connection number of a windmill
graph. In this paper, we shall consider the rainbow connection number of the ducth windmill graphs.
(m)
The dutch windmill graph, denoted Dn , is defined for m ≥ 2 and n ≥ 3 by joining m copies of the(m)(m)
cycle graph Cn with a common vertex. The order of Dn is |V (Dn )| = m(n − 1) + 1 and its size is(m)|E(Dn )| = mn.

(m)Figure 1: D4

for m = 2, 3, 4, 5

82

(m)

A. Suebsriwichai and T. Mouktonglang Rainbow Connection Number of Dn

2

Preliminaries

2.1

Distance and Diameter

TJCDCGGG2021

For a connected graph G, the distance d(u, v) between two vertices u and v is the minimum of lengthsof the u − v paths of G. The diameter diam(G) of G is the maximum of distance between any pair ofvertices. That is,diam(G) = max{d(u, v)|u, v ∈ V (G)}.
If G is a nontrivial connected graph with size |E(G)| and its diameter diam(G), thendiam(G) ≤ rc(G) ≤ |E(G)|

(2.1)
(m)

Lemma 2.1. For each integer m ≥ 2 and n ≥ 3, the diameter of the ducth windmill graph Dnjnk.
diam(Dn(m) ) = 2
2

2.2

is
(2.2)

(m)

Labeling vertices of Dn

(m)

(m)

In this section, we introduce a vertex labeled of Dn for m ≥ 2 and n ≥ 3. Dn consists of m copies(m)of Cn with a common vertex. Given a graph vertex set V (Dn ) = {v0 , v1 , v2 , ..., vm(n−1) }, the common(m)vertex of Dn is defined by v0 . We define each copy of Cn as Cnj where j = 1, 2, 3, ..., m and the labelingvertices in Cnj where j = 1, 2, ..., m to be vi+(j−1)(n−1) for i = 1, 2, ..., n − 1.

3Main Results

Theorem 3.1. For each integer m ≥ 2 and n ≥ 3, the rainbow connection number of the ducth windmill(m)graph Dn isjnk
lnm jnk++ (m − 2)(
− 1).
(3.1)
rc(Dn(m) ) =2
2
2
Acknowledgments. The authors are grateful to the referees for their careful reading of the manuscript
and their useful comments.

References[1] G. Chartrand,G. L. Johns, K. A. McKeon, and P. Zhang, Rainbow connection in graphs, Math. Bohem.,
133, No.2, (2008), 85-98.
[2] Yixiao Liu, Zhiping Wang, The rainbow connection of windmill and corona graph, Applied Mathematical
Sciences, Vol.8, 2014, no.128, 6367-6372.
[3] Syafrizal Sy, Reni Wijaya, and Surahmat Rainbow Connection Numbers of Some Graphs, Applied Mathematical Sciences, Vol. 8, 2014, no. 94, 4693 - 4696.

83

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Rainbow Connection Numbers of s-Overlapping r-Uniform
Hypertrees
S A Farihati1 , A N M Salman2 , and P E Putri2
1

Doctoral Program in Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi
Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia
2
Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut
Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia
sitta@students.itb.ac.id (S A Farihati), msalman@math.itb.ac.id (A N M Salman),etrianaputri@math.itb.ac.id (P E Putri)
Abstract
Let H = (X(H), E(H)) be a non-trivial connected hypergraph. For some k ∈ N, an edge kcoloring of H is a function c : E(H) → {1, 2, . . . , k}. A u-v path in H is said a rainbow, if its edges
have distinct colors. An edge coloring of H is said rainbow connected, if for any two vertices u andv in X(H), there exists a rainbow u-v path. A rainbow connected k-coloring of H is a rainbow
connected coloring of H by using k colors. The rainbow connection number of H, denoted by rc(H),is the smallest natural number k such that there exists a rainbow connected k-coloring of H.
In this talk, we determine the stricts upper and lower bounds for rainbow connection number ofan s-overlapping r-uniform hypertree. For r ≥ 2, r > s ≥ 1, and t ≥ 1, an s-overlapping r-uniform
hypertree is an r-uniform connected hypertree with t edges and every pair of adjacent edges intersects
in precisely s vertices.
Keywords: edge k-coloring, r-uniform hypergraph, rainbow connection number.
2010 MSC: Primary 05C15; Secondary 05C65.

1Introductionr, is an
For r ≥ 2, r > s ≥ 1, and t ≥ 1, an s-overlapping r-uniform hypergraph, denoted by Hs,tr-uniform connected hypergraph with t edges and every pair of adjacent edges intersects in precisely svertices. The two extreme case of s = 1 and s = r − 1 are reffered to as loose and tight hypergraph,
respectively. An s-overlapping r-uniform hypertree is an s-overlapping r-uniform connected hypergraphwhich not contain a cycle. In this talk, we shall consider hypergraphs that are finite, undirected, andsimple (without multiple edges, included edge, or singleton). The terminology and notation used refers
to Berge (1989) [1] and Voloshin (2009) [5].
In 2008, Chartrand et al. [4] introduced the concept of rainbow connection of a graph. Later, thisconcept was adopted by Carpentier et al. [3] to extend the notion of rainbow connection of a hypergraph.
Then Carpentier et al. have implemented it on an r-uniform cycle hypergraph and an r-uniform complete
multipartite hypergraph. In 2019, Budden et al. [2] studied the concept of rainbow connection on an
r-uniform minimally connected hypergraph. In this talk, we consider the rainbow connection number ofs-overlapping r-uniform hypertrees. In particular, we determine the rainbow connection number of ans-overlapping r-uniform interval hypergraph and an s-overlapping r-uniform hyperstar.

2

Rainbow Connections of Hypergraphs

Let H = (X(H), E(H)) be a non-trivial connected hypergraph. For some k ∈ N, an edge k-coloring of H
is a function c : E(H) → {1, 2, . . . , k}. A u-v path in H is said a rainbow, if its edges have distinct colors.
An edge coloring of H is said rainbow connected, if for any two vertices u and v in X(H), there exists a
rainbow u-v path. A rainbow connected k-coloring of H is a rainbow connected coloring of H by using kcolors. The rainbow connection number of H, denoted by rc(H), is the smallest natural number k suchthat there exists a rainbow connected k-coloring of H. As the rainbow connection numbers of a graph,
the rainbow connection numbers of a hypergraph satisfies rc(H) ≥ diam(H).
84

Farihati et al.

Rainbow Connection Number ...

TJCDCGGG2021

3

Main Results

3.1

Rainbow Connection Number of s-Overlapping r-Uniform Interval Hypergraphsr
For r ≥ 2, r > s ≥ 1, and t ≥ 1, an s-overlapping r-uniform interval hypergraph P(s,t)
is an runiform connected interval hypergraph with t edges and every pair of adjacent edges intersects inprecisely s vertices. Therefore, an s-overlapping r-uniform interval hypergraph has the vertex setrr
X(P(s,t)) = {x1 , . . . , xn } with n = (t − 1)(r − s) + r and the edge set E(P(s,t)) = {E1 , E2 , . . . , Et }with Ei = {x(i−1)(r−s)+1 , x(i−1)(r−s)+2 , . . . , x(i−1)(r−s)+r }, 1 ≤ i ≤ t.

Theorem 3.1. Let r ≥ 2, r > s ≥ 1 with r = a mod (r − s) for some a ∈ N, and t ≥ 1, then the rainbowre.
connection number of an s-overlapping r-uniform interval hypergraph is rc(P(s,t)) = d t(r−s)+s−ar−a
For s = r − 1, an s-overlapping r-uniform interval hypergraph is referred to as a tight r-uniform
interval hypergraph with order n, denoted by Pnr . Thus, we have the following.
Corollary 3.2. For n > r ≥ 2, the rainbow connection number of a tight r-uniform interval hypergraphwith order n is rc(Pnr ) = d n−1
r−1 e.

3.2

Rainbow Connection Number of s-Overlapping r-Uniform Hyperstarsr
For r ≥ 2, r > s ≥ 1, and t ≥ 1, an s-overlapping r-uniform hyperstar S(s,t)
is an r-uniform connected
hyperstar with t edges and every pair of adjacent edges intersects in precisely s vertices. Therefore, anrs-overlapping r-uniform hyperstar have the vertex set X(S(s,t)) = {x1 , . . . , xn } with n = t(r − s) + s andr
the edge set E(S(s,t) ) = {E1 , E2 , . . . , Et } with

Ei = {x1 , x2 , . . . , xs } ∪ {xi(r−s)+s , xi(r−s)+(s−1) , xi(r−s)+(s−2) , . . . , xi(r−s)+(s−(r−s−1)) }, 1 ≤ i ≤ t.
Theorem 3.3. Let r ≥ 2, r > s ≥ 1, and t ≥ 1, then the rainbow connection number of an s-overlappingrr-uniform hyperstar is rc(S(s,t)) = t.
For s = r − 1, an s-overlapping r-uniform hyperstar is referred to as a tight r-uniform hyperstar withorder n, denoted by Snr . Thus, we have the following.
Corollary 3.4. For n > r ≥ 2, the rainbow connection number of a tight r-uniform hyperstar with ordern is rc(Snr ) = n − r + 1.

3.3

Rainbow Connection Number of s-Overlapping r-Uniform Hypertrees

The following theorem gives the lower and upper bounds for an s-overlapping r-uniform hypertree. Bothbounds are strict. An example of a hypergraph class that satisfies the lower bound is an s-overlapping
r-uniform interval hypergraph. An example of a hypergraph class that satisfies the upper bound is ans-overlapping r-uniform hyperstar.
Theorem 3.5. Let r ≥ 2, r > s ≥ 1 with r = a mod (r − s) for some a ∈ N, and t ≥ 1, then the rainbow
connection number of an s-overlapping r-uniform hypertree isrd t(r−s)+s−ae ≤ rc(T(s,t)) ≤ t.
r−aMoreover, the lower and upper bounds are sharp.

References[1] C. Berge, Hypergraph : Combinatorics of Finite Set, Elsevier Science Publishers, Netherland, 1989.
[2] M. Budden, J. Hiller, and A. Penland, Minimally connected hypergraphs, arXiv:1901.04560v1 [math.CO](2019).
[3] R.P. Carpentier, H. Liu, Silva, and T. Sousa, Rainbow connection for some families of hypergraphs, Discrete
Mathematics, 327 (2014), 40–50.
[4] G. Chartrand, G.L. Johns, K.A. McKeon, and P. Zhang, Rainbow connection in graphs, Mathematica Bohemica, 133 (2008), 85–98.
[5] V. Voloshin, Introduction to Graph and Hypergraph Theory, Nova Science Publisher, New York, 2009.

85

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Distance to antipode of semi-regular polytope,
measured by edges of equal length
Jin Akiyama1 , and Ikuro Sato2
1

Tokyo University of Science
Miyagi Cancer Centerja@jin-akiyama.com (J. Akiyama), sato-ik510@miyagi-pho.jp (I. Sato)
2

By following the theorem of symmetry, we can tell the number of reflecting hyperplanes of regular andsemi-regular polytopes. This remarkable theorem is derived from the characteristic equation of grouptheory and expressed as Poincare polynomials, defined as a product of Φm+1 (x) = (1 − xm+1 )/(1 − x).
Vertices of semi-regular polytopes locate inside the fundamental domain in primitive or boundary in
non-primitive. Speaking on the primitive class, the distance from a vertex to antipode via edges (edgedistance) is equivalent to the number of reflecting hyperplanes. However, on the non-primitive class, theedge-distance to antipode remains unsolved. In this talk, we will give an algorithm for edge-distance toantipode, not by method of computer searching. This algorithm is based on Wythoﬃan geometry andfeasible both for primitive and for non-primitive.
Our setting problem is oriented toward the edge-distance connecting North Pole and South Pole of a given
semi-regular polytope. It is a matter for argument to avoid an exhaustive searching using a computer
whether the exact distance or its upper bound is feasible or not. This problem may seem to be aneye-measurable property in 3D, but, in a hyperspace, the projection of polytopes failed to be black out
and it is impossible to trace the edges to the antipode. The polytope shown in Fig. 1 belongs to theprimitive class of H4 = [3, 3, 5]. Even though blacken out, the group-theoretical enumeration is possible
and the answer is 60, measured by edges of equal length.

Fig. 1

Table 1

Algorithm for primitives
The edge-distance of primitive is equivalent to the number of reflecting hyperplanes. Because, it is
impossible to arrive at antipode without across every reflecting hyperplane. Here, a set of exponents isdenoted by m = {mi } and Poincare polynomials are defined as a productof Φm+1 (x) = (1−xm+1 )/(1−x).
∑
Edge-distance to antipode is given
as
a
sum
of
exponents,
i.e.,
m
.
Numberof vertices is given as ai∏product of exponents + 1, i.e., (mi + 1). We can consider the exponents as invariants of symmetry andPoincare polynomial as a kind of factorization of discrete space.
An example of A3 = [3, 3]
Truncated octahedron is the primitive of A3 and its generator is written as Poincare polynomial, G(x) =1 + 3x + 5x2 + 6x3 + 5x4 + 3x5 + x6 = (1 + x)(1 + x∏+ x2 )(1 + x + x2 + x3 ) = (1 − x2 )/(1 − x) · (1 −x3 )/(1 − x) · (1 − x4 )/(1 − x) = Φ2 (x)Φ3 (x)Φ4 (x) = Φj (x). The characteristic equation has resulted
in the second kind of Chebyshev polynomial, U3 (x) = 8x3 − 4x = 0, m = {mi } = {1, 2, 3}. Note that86

J. Akiyama et al.

Distance to antipode of semi-regular polytope

TJCDCGGG2021{mi + 1} = {2, 3, 4} reveals double, triple and quadruple rotational symmetry of truncated octahedron.
Φ∑2 (x)Φ3 (x)Φ4 (x) implies factorization of discrete∏ space.
mi = 1 + 2 + 3 = 6 (= n(n + 1)/2 in An ), (mi + 1) = 2 × 3 × 4 = 24 (= (n + 1)! in An ).
An example of B6 = [3, 3, 3, 3, 4]
The characteristic equation has resulted in the first kind of Chebyshev polynomial, T6 (x) = 32x6 −48x4 +2
18x∑ − 1 = 0, m = {mi } = {1, 3, 5, 7,29, 11}. G(x)∏ = Φ2 (x)Φ4 (x)Φ6 (x)Φ8 (x)Φ10 (x)Φ12 (x).
mi = 1+3+5+7+9+11 = 36 (= n in Bn ), (mi +1) = 2×4×6×8×10×12 = 46080 (= 2n · n! in Bn ).
An example of H4 = [3, 3, 5] (Fig. 1/Table 1)
The characteristic equation
has resulted in linear combination of Chebyshev polynomials, 2τ 2∑
T4 (x) −√U4 (x) = 0, τ = (1 + ∏5)/2, m = {mi } = {1, 11, 19, 29}. G(x) = Φ2 (x)Φ12 (x)Φ20 (x)Φ30 (x).
mi =1 + 11 + 19 + 29 = 60, (mi + 1) = 2 × 12 × 20 × 30 = 14400.
Algorithm for non-primitives
For the non-primitive class, the situation is very diﬀerent. Such a powerful calculation for primitives is
not available. Even in diﬃculties, half of the edges are visible and half are invisible. Complemented by
Wythoﬃan geometry, we can tell the edge-distance interpolated for the invisible half.
Upper Bound Theorem: Let Pn , Pn−1 and πn be a semi-regular polytope with Schlaefli-Wythoﬀ code
{p1 , p2 , · · · , pn−1 }(q0 , q1 , · · · , qn−1 ), {p2 , p3 , · · · , pn−1 }(q1 , q2 , · · · , qn−1 ) and {p1 , p2 , · · · , pn−1 }(1, 1, · · · , 1),
respectively. πn stands for the primitive and e(Pn ), e(Pn−1 ) and e(πn ) stand for the edge-distance between North Pole and South Pole. For arbitrary Pn with shape vector q = (q0 , q1 , · · · , qn−1 ), the vectorsv = (v0 , v1 , · · · , vn−1 ) and w = (w0 , w1 , · · · , wn−1 ) are determined corresponding to [p1 , p2 , · · · , pn−1 ].
Then,e(Pn ) ≤ q · v + e(Pn−1 ) ≤ q · w ≤ e(πn ).
An example of lattice n-polytope
When Pn belongs to Bn = [3, 3, · · · , 3, 4], e(πn ) = n2 and e(Pn ) is estimated arithmetically as follows;e(Pn ) ≤ (q0 , q1 , · · · , qn−1 ) · (2, 2, 2, · · · , 2, 1) + e(Pn−1 ) ≤ (q0 , q1 , · · · , qn−1 ) · (2, 4, 6, · · · , 2(n − 1), n) ≤ n2 .
In the case of lattice n-polytope with integer vertex coordinates, edge-distance is exactly given n(n − 1)by a recursive algorithm starting from lattice 3-polyhedron {3, 4}(1, 1, 0) (i.e., 6 = 2 + 4) and stepping
up to higher dimension. Occasionally, the upper bound vectors v and w give us an exact distance.
{3, 3, 4}(1, 1, 1, 0) : q · v = (1, 1, 1, 0) · (2, 2, 2, 1) = 2 + 2 + 2 + 0 = 6, 6 + 6 = 12
{3, 3, 3, 4}(1, 1, 1, 1, 0) : q · v = (1, 1, 1, 1, 0) · (2, 2, 2, 2, 1) = 2 + 2 + 2 + 2 + 0 = 8, 12 + 8 = 20
{3, 3, 3, 3, 4}(1, 1, 1, 1, 1, 0) : q · v = (1, 1, 1, 1, 1, 0) · (2, 2, 2, 2, 2, 1) = 2 + 2 + 2 + 2 + 2 + 0 = 10, 20 + 10 = 30
{3, 3, 4}(1, 1, 1, 0) : q · w = (1, 1, 1, 0) · (2, 4, 6, 4) = 2 + 4 + 6 + 0 = 12 ≤ 16
{3, 3, 3, 4}(1, 1, 1, 1, 0) : q · w = (1, 1, 1, 1, 0) · (2, 4, 6, 8, 5) = 2 + 4 + 6 + 8 + 0 = 20 ≤ 25
{3, 3, 3, 3, 4}(1, 1, 1, 1, 1, 0) : q · w = (1, 1, 1, 1, 1, 0) · (2, 4, 6, 8, 10, 6) = 2 + 4 + 6 + 8 + 10 + 0 = 30 ≤ 36
Conclusion
The setting-up problem has been completed by the above-mentioned algorithm based on Wythoﬃangeometry, not only for primitive class but also for non-primitive class of semi-regular polytopes.

References[1] Coxeter, H. S. M., The Product of the Generators of a Finite Group Generated by Reflections, Duke Math.
Journal 18, 765-782 (1951).
[2] Coxeter, H. S. M., Regular and Semiregular polytopes III, Math Zeit 200, 3-45 (1988).
[3] Ishii M.: On a General Method to Calculate Vertices of N -Dimensional Product-Regular Polytopes, Forma
14, 221-238 (1999)[4] Akiyama J., Ishii M., Sato I. et al., A recursive algorithm for the k-face numbers of Wythoﬃan-n-polytopes
constructed from regular polytopes, J. Information Processing, 25, 528-536 (2017).

87

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai UniversityMöbius Flowers and Buds
Jin Akiyama1 , Kiyoko Matsunaga1 , Sachiko Nakajima1 , and Natsumi Oyamaguchi2
1

Tokyo University of Science, 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601, JAPAN
2
Shumei University, 1-1 Daigaku-cho, Yachiyo, Chiba 276-0003, JAPANja@jin-akiyama.com (Jin Akiyama), matsunaga@mathlab-jp.com (Kiyoko Matsunaga),sachiko.nakajima@steam21.com (Sachiko Nakajima),p-oyamaguchi@mailg.shumei-u.ac.jp (Natsumi Oyamaguchi)
Abstract
If you bisect conjoined two Möbius bands along each centerline, some of them end in interlocking
hearts, and the others end in two separate hearts. What is the cause of this diﬀerence? We would
like to dig deeper to unravel this “Möbius Love-Fate problem” by using the concept of knot theory.
Keywords: Möbius bands, linking number, knot theory
2010 MSC: 97A20

1

Introduction

Let us prepare an elongated rectangle paper. By gluing together the two ends of it, we obtain a band
called an annulus as shown in the upper right figure in Fig.1. If we give one end a half-twist (180 degree
rotation) and glue together the two ends of it, we obtain a twisted band called a Möbius band as shownin the lower right figure in Fig.1. The first thing we notice is that a Möbius band is a non-orientablesurface with only one side and only one boundary. Cutting a Möbius band along the centerline yields
one long band with two full-twists. Note that the length of the resulting band is twice as long as theoriginal Möbius band. We then bisect conjoined two Möbius bands along each centerline, some of themend in interlocking hearts, and the others end in two separate hearts. The question now arises: Whatbrings the diﬀerence between the two? In this article, we would like to reveal this “Möbius Love-Fate
problem” and generalize the number of conjoined Möbius bands to an arbitrary natural number greaterthan 2.

2Preliminaries

There exist two types of Möbius bands depending on the direction of the twist. Here, we provide amethod ∆-check with which we can distinguish them simply. First, we collapse the band into a triangle
shape called a Möbius delta (∆) as shown in the right figure in Fig. 2. There are three sides of thetriangle in layers: the topmost layer (both ends of it are visible), the middle layer (only one end of it isvisible), and the bottom layer (both ends of it are invisible). Label the topmost layer as I, the middle
layer as II, and the bottom layer as III.
A Möbius band is called Type α if the direction of I, II, and III is clockwise. On the other hand,a Möbius band is called Type β if the direction of I, II and III is counter-clockwise. Note that thedirection of I, II, and III, in both types, remains unchanged even if we flip or rotate a Möbius delta.

Fig. 2: ∆-check

Fig. 3: Type α and Type β

Fig. 1: Annulus and Möbius band
In order to consider conjoined N Möbius bands for N ≥ 2, we introduce the following definitions.
88Jin Akiyama et al.

TJCDCGGG2021• Let Ri an elongated rectangle with four vertices Ai , Bi , Ai+N , and Bi+N (i = 0, · · · , N −1) (labeled
clockwisely) as shown in Fig. 4. Note that li is its centerline. Arrange Ri (i = 0, · · · , N − 1) suchthat the clockwisely-rotated l0 around its center by iπ
N coincides with li and glue overlapped partsof Ri as shown in Fig. 5. We call the obtained object an N -star.
• Let us glue together the two ends of Ri of an N -star for each i ∈ {0, 1, · · · , N − 1} and make N
annuli Ai so that Ai+1 goes over Ai . We call the obtained object an N -bud(Fig. 6). Note that
Ai is attached to Bi+N and Bi is attached to Ai+N .
• Let us glue together the two ends of Ri of an N -star for each i ∈ {0, 1, · · · , N − 1} and make N
Möbius bands Mi so that Mi+1 goes over Mi . We call the obtained object an N -flower(Fig.
7). Note that Ai is attached to Ai+N and Bi is attached to Bi+N . Let the flower type S be asequence of the type of Möbius bands Mi for i ∈ {0, 1, · · · , N − 1}. (e.g., S = α · · · α or αβα · · · α)Note that if we bisect all the Möbius bands Mi (i = 0, · · · , N − 1) in N -flower, N -flower ends in N
(heart-shaped) bands whether they are interlocked or separated. Let Pi denote one of these bands which
includes Ai , Ai+N , Bi−1+N , and Bi−1 and call it a petal for each i ∈ {0, 1, · · · , N − 1}. We note thatthe subscripts are congruent modulo 2N .

Fig. 4: Rectangle Ri

3

Fig. 5: N -starFig. 6: 4-bud

Fig. 7: 5-flower

Main Results

Theorem 3.1.
1. (Möbius Love-Fate Theorem) Let N = 2. If the flower type S = αα or ββ, twopetals P0 and P1 are separated. If the flower type S = αβ or βα, they are interlocked. (Fig. 8)
2. (N-flower Theorem) Let N be an integer greater than 2. If Mi (i ∈ {0, · · · , N − 1}) is type α,
petals Pi and Pi+1 are interlocked. If Mi (i ∈ {0, · · · , N − 1}) is type β, petals Pi and Pi+1 areseparated. Pi and Pj are separated if |i − j| > 1. Especially, a heart ring is obtained from theflower type S = αα · · · α, while, a heart chain is obtained from the flower type S = αα · · · αβ. If
the flower type is S = ββ · · · β, the mutually-separated N hearts are obtained. (Fig. 9)Note that the Olympic logo-like heart chain will appear from the 5-flower of type βαααα. (Fig. 10)

Fig. 8: Möbius Love-Fate

Fig. 10: Olympic logo-like heart chain

Fig. 9: N petals

Theorem 3.2.
1. Let N be an even number. If we bisect each annulus Ai (i = 0, · · · , N − 1) of a
N-bud along its center line, a long band, whose length is 2N times as long as the original annulus,
is obtained.(Fig. 11) Especially, a square frame is obtained from a 2-bud.(Fig. 12)
2. Let N be an odd number. If we bisect each annulus Ai (i = 0, · · · , N − 1) of a N-bud along its
center line, the interlocked two long bands, whose lengths are both N times as long as the originalannulus, are obtained.(Fig. 13)

Fig. 11: A long band from 4-bud

Fig. 12: A square frame from 2-bud Fig. 13: Long bands from 3-bud89

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

The Conditions for Fitting a Square in a Trapezoid
Nattaphon Rintaew1,2 , Supanut Chaidee1
1

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Chiengrai Vidhayakhome School, Chiang Rai, Thailand
nattaphonrintaew@gmail.com (N. Rintaew), supanut.c@cmu.ac.th (S. Chaidee)
2

Abstract
We propose the necessary and suﬀicient conditions which fit a square in a trapezoid when the
trapezoid is cut from a triangle parallel to its base. The conditions are presented in the form of thelargest fitting square on each trapezoid side when the original triangle is right and acute triangle
such that the sum of base angle is greater than π/2.
Keywords: largest inscribed square, polygon containment, trapezoid.
2010 MSC: Primary 52Cxx; Secondary 52C15.

1 Introduction
Assume that a polygon is given, the problem to find the necessary and suﬀicient conditions which otherpolygon fits in it were widely studied as shown in the previous studies. Wetzel [7] compiled the fitting
and covering problems, and many variations were investigated such as [6] and [5]. In the computational
geometry viewpoints, these problems were considered as the polygon containment problem as compiledin [3]. Agarwal et al. [1] proposed an algorithm to find the largest inscribed equilateral triangles andsquares in convex and arbitrary polygons.
In this study, we propose the necessary and suﬀicient conditions when a square is fit in a trapezoidby cutting a triangle parallel to the base of a triangle. The condition as presented in [6] will be applied
to each case of trapezoid obtained from cutting off the triangle.

2 Preliminaries
Assume that a trapezoid T has the height d, the length of parallel edges d1 , d2 such that d1 < d2 , andits base angle are α, β. Remark that a trapezoid obtaining by cutting a triangle parallel to its basecan be separate into 4 types in Figure 1. In this study, we focus on the case α = π/2, β < π/2 andα < π/2, β < π/2 such that α + β ≥ π/2.

Figure 1: characterization of trapezoids with their parameters
Theorem 2.1 considers the properties of the largest square in a convex polygon P . When the convex
polygon P is a triangle, the basic conditions were proposed by Brune [2] in Theorem 2.2. Wetzel finallyproposed the necessary and suﬀicient conditions in [6]. The theorems based on the acute and right angle
triangle are mentioned as follows.
90

N.Rintaew et al.

The Conditions for Fitting a Square in a Trapezoid

TJCDCGGG2021

Theorem 2.1. [1] A maximum area square inscribed in a convex polygon P must satisfy at least oneof the following two conditions: At least one corner is coincident with a vertex of P , or all four corners
lie on the interior of edges of P .
Theorem 2.2. [2] A triangle T is given. If T is acute, the largest inscribed square has a side on
the shortest of T ; if T is right, the inscribed square on the two legs is larger square with a side on the
hypotenuse; and if T is obtuse, there is only one inscribed square, which is ipsofacto largest.
Theorem 2.3. [6] Let ABC be an acute or right triangle such that c is a shortest side of T . Then asquare with side s fits in T if and only if s ≤ 2∆c/(c2 + 2∆), where ∆ is the area of T .

3 Main Results
When a trapezoid T is cut from the right triangle, the conditions can be separated into two parts: thelargest square satisfying the condition of [6] in the case of a right triangle, and the cut below the conditionof [6]. If T is derived from the right angle T or acute isosceles, we obtain the following condition.
d + d1 tan β
when
Theorem 3.1. Let x be the length of a square S. Then S can be fit in T iff x ≤1 + tan βd1 < d, and x ≤ d when d ≤ d1 .
When a triangle ABC is an acute triangle such that a is the smallest length, Theorem 2.2 implies
that there is a length σ which indicates the condition when we cut the triangle as shown in Figure 2.

Figure 2: Orientation of largest inscribed square in a triangle ABC
Theorem 3.2. Let x be the length of the inscribed square in an acute triangle T , and xI be the lengthof a square S. Then S can be fit in T if it satisfies the following conditions.
d2 sin α(cot α − cot(α + β))
1. When d > x sin α + x cos α, xI ≤.
1 + cot α − cot(α + β)
2. When d ≤ x sin α + x cos α, we consider the following cases:




1
1
1
1
(a) when d1 < d+−1 +,tan α tan β
sin α + cos α
sin α + cos α
xI ≤


1
1
+
tan α tan β(c) when d ≤ d1 , xI ≤ d.
(b) when d1 ≥ dd tan α + d tan β + d1 tan α tan β;
tan α + tan β + tan α tan β



v

u
u 1−tan α 2 + 1
t
1+tan α1
1;
−1 +, xI ≤ dsin α + cos α
sin α + cos α2References[1] Agarwal, P. K., Amenta, N., & Sharir, M. (1998). Largest placement of one convex polygon inside another.
Discrete & Computational Geometry, 19(1), 95-104.
[2] Brune, Grofstes Quadrat im Dreiecke, Crelle’s Journal (= J. reine angew. Math.) 15 (1836) pp. 365-366[3] Chazelle, B. (1981). The polygon containment problem. Carnegie-Mellon University, Department of Computer
Science[4] DePano, A., Ke, Y., & O’Rourke, J. (1987). Finding largest inscribed equilateral triangles and squares. In
Proc. 25th Allerton Conf. Commun. Control Comput (pp. 869-878).
[5] Wetzel, J. E. (2000). Rectangles in rectangles. Mathematics Magazine, 73(3), 204-211.
[6] Wetzel, J. E. (2002). Squares in triangles. The Mathematical Gazette, 86(505), 28-34.
[7] Wetzel, J. E. (2003). Fits and covers. Mathematics magazine, 76(5), 349.

91

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Previous Player’s Positions of Impartial Three-Dimensional
Chocolate-Bar Games
Ryohei Miyadera1 and Hikaru Manabe 1
1

Keimei Gakuin Junior and High School, Kobe City, Japanrunnerskg@gmail.com (R. Miyadera), urakihebanam@gmail.com (H. Manabe)
Abstract
The authors presented research on three-dimensional chocolate games that are variants of CHOMP.
A three-dimensional chocolate bar is a three-dimensional array of cubes in which a bitter cubic box
exists in some parts of the bar. Two players take turns and cut the bar horizontally or verticallyalong the grooves. The player who manages to leave the opponent with a single bitter block is thewinner. The authors studied the P-positions of chocolate games, where P-positions are positions of
the game from which the previous player (the player who will play after the next player) can force awin, provided he/she plays correctly at every stage. The set of P-positions is important because it
provides the winning strategy of the game. The authors discovered some suﬃcient conditions for the
chocolate bars by which a position {p, q, r} is a P-position if and only if (p − 1) ⊕ (q − 1) ⊕ (r − 1),
where p, q, and r are the length, height, and width of the chocolate bar, respectively.
Keywords: Combinatorial game, chocolate game, previous player s position, CHOMP
2010 MSC: Primary 91A46; Secondary 97A20.

1Introduction

In this study, the authors present a three-dimensional chocolate game that consists of a three-dimensional
array of cubes in which a bitter cubic box printed in black exists in some parts of the bar. The shape of
chocolate bar is deﬁned by 1.1 using the coordinate system that shows the size of the chocolate bar in
Fig. 2. This game is played under the rule in Deﬁnition 1.1. Fig. 1presents three methods of cutting athree-dimensional chocolate bar. This is a consequence of the research presented in [1] and [2].
Deﬁnition 1.1. (i) Let x, y, z ∈ Z≥0 and f (x, z) be a monotonically increasing function with respect tox, z. The three-dimensional chocolate bar comprises a set of 1 × 1 × 1 boxes. For u, w ∈ Z≥0 , such thatu ≤ x and w ≤ z, the height of the column at position (u, w) is y + 1, where y ≤ f (u, w). In other words,
the maximum value of the height is always equal to or under f (u, w) + 1, and hence the maximum value
depends on u and w. A unique bitter box printed in black is placed at position (0, 0). We denote thischocolate bar as CB(f, x, y, z).
Note that x + 1, y + 1, and z + 1 are the length, height, and width of the bar, respectively.
(ii) Each player, in turn, cuts the bar horizontally or vertically along the grooves and eats the broken
piece. The player who manages to leave the opponent with a single bitter block (black block) is thewinner.

5
4
5

3

y

x 3

2

4

2

1

1

0 0
0

1

z

Figure 1.

2

3

4

5

Figure 2.

Figure 3.

The chocolate bar in Fig. 3 is CB(f, 9, 5, 7), where f (x, z) = ⌊ x+z
3 ⌋.
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Previous Player’s Positions of Chocolate GamesTJCDCGGG2021∑n
Deﬁnition 1.2. Let x and y be nonnegative integers. We express x and y in base 2 as x = i=0 xi 2in∑n∑
and y = i=0 yi 2i with xi , yi ∈ {0, 1} and n ∈ N . We deﬁne nim-sum x ⊕ y as x ⊕ y =wi 2i , wherei=0

wi = xi + yi (mod 2).
For completeness, we brieﬂy review some of the necessary concepts of combinatorial game theory;refer to [3] for more details. As chocolate bar games are impartial games without draws, only two
outcome classes are possible.
Deﬁnition 1.3. (a) A position is called a P-position, if it is a winning position for the previous player(the player who just moved), as long as he/she plays correctly at every stage.
(b) A position is called an N -position, if it is a winning position for the next player, as long as he/she
plays correctly at every stage.
Deﬁnition 1.4. (i) For any position p of game G, there is a set of positions that can be reached by
precisely one move in G, which we denote as move(p).
(ii) The minimum excluded value (mex) of a set S of nonnegative integers is the least non-negative
integer that is not in S.
(iii) Each position p of an impartial game has an associated Grundy number, and we denote it as G(p).
The Grundy number is found recursively: G(p) = mex({G(h) : h ∈ move(p)}).
Theorem 1.5. For any position p of the game, G(p) = 0 if and only if p is a P-position.

2

Results

One of the most important topics of research on combinatorial games is to obtain a closed formula for
Grundy numbers. The authors studied the necessary and suﬃcient conditions for the shape of chocolatebars in which the Grundy number of chocolate bar is equal to (p − 1) ⊕ (q − 1) ⊕ (r − 1), where p, q,
and r are the length, height, and width of the chocolate bar, respectively. They published their resultson two-dimensional chocolate bar in [2], and currently, they have already submitted the result in athree-dimensional case to a journal.
Suppose that the Grundy number of the chocolate bar is (p − 1) ⊕ (q − 1) ⊕ (r − 1), where p, q, and r
are the length, height, and width of the chocolate bar, respectively. Then, by Theorem 1.5, a chocolate
bar is a P-position if and only if (p − 1) ⊕ (q − 1) ⊕ (r − 1) = 0.
Therefore, it is natural to ﬁnd the condition under which a three-dimensional chocolate bar withlength p, height q, and width r is P-position if and only if (p − 1) ⊕ (q − 1) ⊕ (r − 1) = 0, although it isdiﬃcult because there are many varieties of chocolate shapes that satisfy this condition. However, theauthors discovered some suﬃcient conditions.
Theorem 2.1. Let f (x, z) = ⌊ x+z
k ⌋ for k = 4m + 3. Then chocolate bar CB(f, x, y, z) is a P-position
if and only if x ⊕ y ⊕ z = 0.
Theorem 2.2. Let f (x, z) = ⌊ x+z
k ⌋ for k = 8m + 4 with m ∈ Z≥0 . Then, under the condition thatx, y, z ≤ 12m + 7, chocolate bar CB(f, x, y, z) is a P-position if and only if x ⊕ y ⊕ z = 0.
The authors also obtained the following two conjectures that are derived by calculation of computer
algebra system Mathematica.
Conjecture 2.3. Let f (x, z) = ⌊ x+z
k ⌋ for k = 4m + 1. Therefore, chocolate bar CB(f, x, y, z) is a
P-position if and only if (x + 1) ⊕ y ⊕ (z + 1) = 0.
Conjecture 2.4. Let f (x, z) = ⌊ x+z
k ⌋ for k = 16m+8. Then, under the condition that x, y, z ≤ 56m+31,
chocolate bar CB(f, x, y, z) is a P-position if and only if x ⊕ y ⊕ z = 0.
If we compare these theorems and conjectures, we ﬁnd that the case of f (x, z) = ⌊ x+z
k ⌋ with an odd
number k is diﬀerent from the case with an even number k. In the case of an even number k, there is a
restriction on the size of the chocolate bar.

References[1] Y. Nakaya, M. Fukui, R. Miyadera, S.Nakamura Grundy Numbers of Impartial Three Dimensional Chocolate
Bar Games, JCDCG3 (2016).
[2] S. Nakamura, R. Miyadera and Y. Nakaya, Impartial Chocolate Bar Games, Grundy Numbers of Impartial
Chocolate Bar Games, Integers Vol. 20, (2020), #G1.
[3] M. H. Albert, R. J. Nowakowski and D. Wolfe, Lessons in Play, A K Peters, (2019).

93Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Chocolate Games and Restricted Nim
Ryohei Miyadera1 , Hikaru Manabe 1 Kousei Suzuki 1 , Taishi Aono1 , Shouei Takahasi1 and Sohta
Kannan2
1

Keimei Gakuin Junior and Senior High School, Kobe City, Japan

2

Schools of Science and Technology, Kwansei Gakuin University, Sanda City, Japanrunnerskg@gmail.com (R. Miyadera), urakihebanam@gmail.com (H. Manabe)
Abstract

In this study, we present the results of research on rectangular chocolate games with a restriction
on the size of the chocolate piece to be eaten. The first result involves a closed formula for calculating
the previous player’s winning position. In this research, we used the results of our previous researchon Max Nim presented at JCDCG3 2018. The second result is the research on rectangular chocolatebars with some parts are missing. The values of the Grundy number G(x, y) depend on the locationand size of the missing parts.
Keywords: Combinatorial game, chocolate game, previous player’s position, restricted nim
2010 MSC: Primary 91A46; Secondary 97A20.

1Introduction

We herein studied a chocolate game with a restriction on the size of the chocolate piece to be eaten andpresented closed formulas for the previous player’s winning position. In this research, we used the resultsof their previous research on Max Nim presented in [1]. Let Z≥0 be the set of non-negative numbers.
Definition 1.1. Let g(n) ∈ Z≥0 be a monotonically increasing function defined on Z≥0 .
(i) We obtain the coordinate system shown in Fig.2. In this example, we describe this chocolate as {5, 3}.
(ii) A chocolate bar is a rectangular array of squares, and there is a bitter square at the bottom of thebar. Two players take turns. When a player’s turn arrives, she or he must break the chocolate bar along
any one of the horizontal or vertical lines into two pieces and eat the piece without the bitter part. As
for the chocolate with the size of {x, y}, when one of the players eats the piece without a bitter part,what remains is {x − t, y}-chocolate with t ≤ g(x) or {x, y − t}-chocolate with t ≤ g(y).
There are diﬀerent types of chocolate bars (see Fig.1). The authors have studied chocolate bars withsome squares removed without any restriction on the size of the chocolate bar to be eaten, and they
published their research in [2].

Figure 1.

Figure 2.

Figure 3.

We assume that g(n) = ⌈ nd ⌉ with a real number d such that 1 < d. For example, if d = 1.5, then by(ii) of Definition 1.1, you get one of the chocolates shown in Fig. 3 when you start with the one in Fig.
2.
Definition 1.2. (a) P-positions, that is, the winning positions for the previous player (the player whojust moved), as long as they play correctly.
(b) N -positions, that is, the winning positions for the next player, as long as they play correctly.
∑n∑nDefinition 1.3. Let x, y ∈ Z≥0 . We represent x = i=0 xi 2i and y = i=0 yi 2i with xi , yi ∈ {0, 1}.
n∑
We define the nim-sum x ⊕ y by x ⊕ y =wi 2i , where wi = xi + yi (mod 2).
i=094

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Chocolate Games and Restricted Nim

TJCDCGGG2021

Definition 1.4. (i) For any position p of a game G, there is a set of positions that can be reached bymaking precisely one move in G; this is denoted by move(p). In the game of Definition 1.1,
move({x, y}) = {{x − t, y} : t ≤ g(x)} ∪{{x, y − t} : t ≤ g(y)}.
(ii) The minimum excluded value (mex) of a set S of non-negative integers is the least non-negative
integer not in S.
(iii) Let p be the position of an impartial game. The associated Grundy number is denoted by G(p)and is recursively defined by G(p) = mex{G(h) : h ∈ move(p)}.
Theorem 1.5. Let G be the Grundy number. Then, a position p is a P-position if and only if G(p) = 0．

2

Main Results

We define a restricted nim.
Definition 2.1. Suppose there is a pile of n stones, and two players take turns to remove stones fromthe pile. In each turn, if the number of stones is m, the player is allowed to remove at least one and at
most g(m) stones, where g(m) = ⌈ md ⌉ with a real number d such that 1 < d. The player who removes
the last stone or stones is the winner.
We let G1 (x) be the Grundy number of the restricted nim of Definition 2.1 when there are x-stones,and G2 (x, y) is the number of chocolate games in Definition 1.1 when the size of the chocolate is {x, y}.
Theorem 2.2. Suppose that 1 < d. Then, there exists a non-negative integer valued function hd with aclosed formula such that G1 (hd (n)) = G1 (n) for n ∈ Z≥0 .
Theorem 2.3. G2 (x, y) = G1 (x) ⊕ G1 (x).
This is clear because the game in Definition 1.1 is the sum of the two games in Definition 2.1.
Theorem 2.4. Then, the set of P-positions is {{hpd (n), hqd (n)} : n ∈ Z≥0 and p, q ∈ N }.
This theorem follows from theorems 1.5 2.2 and 2.3.
Next, we study a chocolate game with the condition that the piece to be eaten should be equal toor smaller than the remaining piece, and chocolates are any one of the chocolates in Fig. 4. Here, thechocolates are without bitter parts, except for the fifth one in Fig. 4.
120

100

80

60

40

Figure 4. A chocolate without missing parts, a chocolate with one missing part, a chocolate with two missing parts, a chocolate with three missing parts, and achocolate with three bitter parts.

20

20

40

60

80

100

120

140

Figure 5. fractal pattern described in conjecture 2.6.

Let Gk (x, y) be the Grundy number of rectangular chocolates with k-missing parts for k = 0, 1, 2, 3,
where x, y are the width and height of the chocolate.
Theorem 2.5. We have the following (i) and (ii).
(i) G1 (x, y) = G0 (x, y).
(ii) G2 (x, y) = G0 (x, y) if y ≥ 3 and (x, y) ̸= (3, 3).
The following conjecture is based on the calculation by computer algebra system Mathematica.
Conjecture 2.6. (i) G3 (x, y) = G0 (x, y) for (x, y), such that y ≥ x, and the set {(x, y) : G3 (x, y) ̸=G0 (x, y)} has a fractal structure, as shown in the graph in Fig. 5.
(ii) Grundy numbers of the fifth chocolate game in Fig. 4 are similar to Grundy numbers of the first
chocolate game.

References[1] R. Miyadera et al., Some Formulas for Max Nim, JCDCG3 2018, Manila, pp.35–36, August, 2018.
[2] S. Nakamura, R. Miyadera and Y. Nakaya, Impartial Chocolate Bar Games, Grundy Numbers of Impartial
Chocolate Bar Games, Integers Vol. 20, (2020).

95

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Permutation-Generated Maps between Dyck Paths
Kevin Limanta1 , Hopein Christofen Tang2 , and Yozef Tjandra3
1

School of Mathematics and Statistics, UNSW Sydney, NSW 2052, Australia
Department of Mathematics, Institut Teknologi Bandung, Bandung 40132, Indonesia
3
Department of IT & Big Data Analytics, Calvin Institute of Technology, Jakarta 10610, Indonesia
k.limanta@unsw.edu.au (K. Limanta), hopeinct@students.itb.ac.id (H. C. Tang),yozef.tjandra@calvin.ac.id (Y. Tjandra)
2

Abstract
In 2003, Deutsch and Elizalde defined a family of bijections between the set of Dyck paths toitself which is induced by some particular permutations. In this paper, we extend the construction of
the maps by allowing the permutation to be arbitrary and explore some consequences. Of particularinterest, this leads to a reformulation of an existing combinatorial identity, a characterisation of
bijection-generating permutations, and an equidistribution between some statistics of Dyck paths tothe height statistics.
Keywords: Dyck paths, bijection on Dyck paths, statistic on Dyck paths.
2010 MSC: Primary 05A05; Secondary 05A18, 05A19.

1 Introduction
Bijective method is one of the classic tools in enumerative combinatorics. One enumeration problem canbe translated to another, often in different discrete structure. In particular, we put a certain interest on
the bijections between Dyck paths defined by Deutsch and Elizalde [1] having application in enumerating
Dyck paths with some specific statistics as well as pattern-avoiding permutations restricted to several
rules.
In this work, we muse on a natural generalisation of the maps defined by Deutsch and Elizalde byletting the maps be generated by arbitrary permutations. Motivated by the success of Deutsch andElizalde’s bijections, this generalisation, as we shall see, gives rise to new family of bijections whichmight be useful in the combinatorics of Dyck paths.

2 Preliminaries
2.1 Dyck Paths
A Dyck path of semilength n is a lattice path between (0, 0) and (2n, 0) consisting of up-steps (1, 1)
and down-steps (1, −1), represented by u and d consecutively, which never goes below the x-axis. Stepk of a Dyck path P is denoted by Pk ∈ {u, d}. Let Dn denote the set of all Dyck paths of semilength n.

Figure 1: A Dyck path P = uuduuddd, P2 = u, and P7 = d. The dashed lines denote the tunnel pairings.
For any Dyck path D, define a tunnel of D to be the horizontal segment between two lattice pointsof D that intersects D only in those two points and always stays below D. If D ∈ Dn , then D has
exactly n tunnels and each tunnel can be associated with a pair of (k, l) such that step k and step l areconnected by a tunnel. The tunnel pairs of D can be encoded as a permutation τD , called the tunnelingof D, where τD (k) = ℓ if and only if (k, ℓ) is a tunnel pair.
96

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TJCDCGGG2021

2.2 Permutation Generated Maps between Dyck Paths
Definition 2.1. Given a permutation σ ∈ S2n and a Dyck path D, the σ-path of D, denoted by σ(D),is a lattice path constructed by the following algorithm: at iteration k ∈ [2n],(u if τD (σk ) ∈/ {σ1 , σ2 , . . . , σk },σ(D)k =
d otherwise.
The map D 7→ σ(D) is denoted by σ(·).
The original map defined by Deutsch and Elizalde in [1, Section 3] takes a particular choice of
permutation σ which goes in zig-zag, that is(
k+1
if k is odd,
2σk =2n + 1 − k2 if k is even.

3 Main Results
By noticing that some different permutations could generate an exact same map, a certain equivalence
relation related to the permutation-generated maps can be utilised to rediscover an existing combinatorial
identity connecting a certain statistic of Dyck paths and the double factorial term, as presented below.
Proposition 3.1 (Proposition 1 of [2]). Let P be a Dyck path of size n and huk be the height of stepuk ∈ UP . We have thatn
X Y
huk (P ) = (2n − 1)!!.
P ∈Dn k=1

The following theorem characterises permutations which generate bijections. A permutation σ ∈ S2n
is called a circularly connected permutation (CCP) if in a cycle graph Cn whose vertices are labelledfrom {1, 2, . . . , 2n}, the subgraph induced by {σ1 , σ2 , . . . , σk } is always connected for all k ∈ [2n].
Theorem 3.2. σ(·) is a bijection if and only if σ is a CCP.
A particular application of the permutation-generated bijections is to show the equidistribution ofcertain statistics of Dyck paths depicted as the following theorem. Suppose that P is a Dyck path. Letua,k (P ) denote the number of steps in Ua,k = {Pa , Pa+1 , . . . , Pa+k−1 } (with the indices arithmetics is(a)done under modulo 2n) whose tunnel pair is not in Ua,k . Define umax (P ) = max ua,k (P ). Define alsok∈[2n]

h(P ) to be the height of the highest peak of P .
Theorem 3.3. Let a, k ∈ [2n] be arbitrary. We have that for any ℓ ∈ [n],{P ∈ Dn : ua,k (P ) = ℓ} = {P ∈ Dn : hk (P ) = ℓ} ,as well as{P ∈ Dn : u(a)max (P ) = ℓ} = {P ∈ Dn : h(P ) = ℓ} .

References[1] Elizalde S. and Deutsch E. “A simple and unusual bijection for Dyck paths and its consequences”. In: Annalsof Combinatorics 7.3 (2003), pp. 281–297.
[2] Callan D. “A combinatorial survey of identities for the double factorial.” arXiv preprint arXiv:0906.1317(2009).

97

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Dynamic Independent Set of Squares
Sujoy Bhore1 , Jean Cardinal2 , John Iacono2 , and Grigorios Koumoutsos2
1

Indian Institute of Science Education and Research, Bhopal, India
Université libre de Bruxelles, Belgium.
sujoy.bhore@gmail.com (S. Bhore), jean.cardinal@ulb.be (J. Cardinal), john@johniacono.com (J. Iacono),
greg.koumoutsos@gmail.com (G. Koumoutsos)
2

Abstract
We present fully dynamic approximation algorithms for maintaining the maximum independent
set of axis-aligned squares and hypercubes. It is known that a maximum independent set of acollection of n intervals can be found in O(n log n) time, while it is NP-hard already for a set of unit
squares. The problem is inapproximable on many important graph families, however, admits a PTAS
for a set of arbitrary pseudo-disks. A fundamental question in computational geometry is whether
it is possible to maintain an approximate maximum independent set for a set of dynamic geometricobjects, in truly sublinear time per insertion or deletion. In this work, we answer this question in
the affirmative for squares and hypercubes. Our main result is a data structure for maintaining an
expected constant factor approximate maximum independent set of axis-aligned squares in the plane,
with polylogarithmic amortized update time. Moreover, our approach generalizes to d-dimensional
hypercubes, providing a O(4d )-approximation with polylogarithmic update time. These are the first
fully dynamic approximation algorithms for any set of dynamic arbitrary size squares and hypercubes;
previous results required bounded size ratios to obtain polylogarithmic update time. It is known that
our results for squares (and hypercubes) cannot be improved to a (1 + ε)-approximation with thesame update time.
Keywords: Geometric independent sets, dynamic algorithms.
2010 MSC: Computational geometry; Dynamic graph algorithms.

1

Introduction

In the maximum independent set (MIS) problem, we are given a graph G = (V, E) and the objective
to find a subset I ⊆ V of maximum cardinality, such that no two vertices in I are adjacent. MIS is afundamental graph problem with a wide range of applications, and is one of Karp’s 21 classic NP-complete
problem. The problem is also known to be hard to approximate on general graphs: no polynomial timealgorithm can achieve an approximation factor n1−ε , for any constant ε > 0, unless P=NP.
Geometric Independent Set. In this work, we focus on geometric setting, called geometric independent sets. In this setting, we are given a set S of geometric objects, and the graph G is their intersection
graph, where each vertex corresponds to an object, and two vertices form an edge if and only if thecorresponding objects intersect. Independent sets of geometric objects in the plane such as axis-aligned
squares or rectangles have been extensively studied due to their various applications in e.g., VLSI design,
map labeling and data mining. However, even the case of computing a MIS of axis-aligned unit squares
is known to be NP-complete. On the positive side, several geometric cases including pseudo-disks, admit a polynomial time approximation scheme (PTAS); see [2]. Recently, Gálvez et al. [3] obtained aconstant-factor approximation algorithm for the MIS problem on axis-parallel rectangles in the plane.
Dynamic Geometric Independent Set. In the dynamic version of the geometric independent Set
problem, objects are inserted and deleted over time. The goal is to achieve (almost) the same approximation ratio as in the offline (static) case while keeping the update time, i.e., the running time required tocompute the new solution after insertion/deletion, as small as possible. Henzinger et al. [1] studied thegeometric independent set problem for intervals, hypercubes and hyperrectangles. They obtained several
results, many of which extend to the substantially more general weighted independent set problem where98

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Dynamic Independent Set of Squares

TJCDCGGG2021

objects have weights. However, to obtain the upper bounds, they considered the special case where all
objects are located in [0, N ]d and have minimum length edge of 1, hence also bounded size ratio of N ,and presented dynamic algorithms with approximation ratio (1 + ε) for intervals and (1 + ε) · 2d for
d-dimensional hypercubes with update time polylog(n, N ). We note that in general, N might be quite
large such as exponential in n or even unbounded, thus those bounds are not truly sublinear in n in thegeneral case. In [4], we showed that it possible to maintain a (1 + ε)-approximate independent set ofintervals under insertions and deletions of intervals, in O( logε n ) worst-case update time per insertion andnO( logε2 ) per deletion, where ε > 0 is any positive constant and n is the total number of intervals. Thisis the first algorithm yielding such a guarantee in the comparison model, in which the only operations
allowed on the input are comparisons between endpoints of intervals. Later, Compton et al. [5] improved
this result by designing an (1 + ε)-approximate dynamic algorithm which supports both insertions anddeletions in time O( logε n ).

1.1

Our Contribution

In this work, we consider the problem of maintaining dynamically an independent set of squares andhypercubes. In [1], Henzinger et al. established lower bounds that implies that no PTAS can be achievedin polylogarithmic time. Hence we turn our attention to a fully dynamic constant factor approximation
algorithm. We prove the following theorem.
Theorem 1.1. There exist algorithms for maintaining an expected O(1)-approximate independent set ofaxis-aligned squares in the plane under insertions and deletions of squares, in O(log5 n) amortized time
per update, where n is the total number of squares.
In what follows, we provide a very high-level description. The detailed proof can be found in the long
version of the paper [4].
High-level description. Our approach to maintain an approximate MIS of squares is divided into twoparts: static construction and dynamization. We define a randomly scaled and shifted infinite quadtree,
and associate each square with the smallest enclosing node of the quadtree. We call the squares thatintersect the center of their quadtree node centered and discard all noncentered squares. Nodes of
the quadtree associated with squares are called the marked nodes of the infinite quadtree, and let thequadtree Q be the union of all the marked nodes and their ancestors. Note that multiple squares may be
associated with one quadtree node. The size of Q is unbounded but the total number of marked nodes isO(n). We show that by losing a factor of 16 in expectation, thus we can restrict our attention to centeredsquares, thus we can indeed discard all non-centered squares. Then, we reduce the case of squares tointervals using the compressed random quadtree and decompose it carefully into relevant paths, andshow that using an optimal solution for intervals we can obtain a O(1)-approximate solution for squares
in polylogarithmic time. However, to make this structure dynamic, more involved work is needed due tothe changing nature of the quadtree as well as to incorporate an approach used for approximate dynamic
independent intervals.
Hypercubes and Beyond We show that our approach for squares naturally extends to axis-aligned
hypercubes in d dimensions, providing a O(4d )-approximate independent set in O(2d log2d+1 n) time.
Acknowledgments. S. B. and J. I. are supported by the Fonds de la Recherche Scientifique-FNRS
under Grant no MISU F 6001 1. G.K is supported by FNRS under a “Scientific Collaborator” grant.

References[1] M. Henzinger and S. Neumann and A. Wiese, Dynamic Approximate Maximum Independent Set of Intervals,
Hypercubes and Hyperrectangles, In the proceedings of SoCG, 2020, June 23–26, pp. 51:1–51:14.
[2] T. M. Chan and S. Har-Peled, Approximation Algorithms for Maximum Independent Set of Pseudo-Disks,
Discrete & Computational Geometry 48 (2012), 373–392.
[3] W. Gálvez and A. Khan and M. Mari and T. Mömke and M. Reddy Pittu, A 4-Approximation Algorithm
for Maximum Independent Set of Rectangles, arXiv abs/2106.00623 (2021).
[4] S. Bhore and J. Cardinal and J. Iacono and G. Koumoutsos, Dynamic Geometric Independent Set, arXiv
abs/2007.08643 (2020).
[5] S. Compton and S. Mitrovic and R. Rubinfeld, New Partitioning Techniques and Faster Algorithms for
Approximate Interval Scheduling, arXiv abs/2012.15002 (2020).

99

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Γ-Induced-Paired Dominating Graphs of Cycles
Saharath Sanguanpong and Nantapath Trakultraipruk
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University,
Pathumthani 12120, Thailand
shr.sanguanpong@gmail.com (S. Sanguanpong), n.trakultraipruk@yahoo.com (N. Trakultraipruk)
Abstract
A set D of vertices of a graph G is an induced-paired dominating set if the subgraph of G induced
by D contains only nonadjacent edges and every vertex of G is adjacent to some vertex in D. The
set D is minimal if every proper subset of D is not an induced-paired dominating set. The upper
induced-paired domination number of G, denoted by Γip (G), is the maximum cardinality of a minimal
induced-paired dominating set of G. An Γip (G)-set is a minimal induced-paired dominating set of
cardinality Γip (G) . The Γ-induced-paired dominating graph of a graph G, denoted by IP DΓ (G), is
the graph whose vertex set is the set of all Γip (G)-sets. Two Γip (G)-sets D1 and D2 are adjacent in
IP DΓ (G) if D2 = D1 \{v1 } ∪ {v2 } for some v1 ∈ D1 and v2 ∈/ D1 . In this paper, the Γ-induced-paired
dominating graphs are determined for all cycles.
Keywords: Induced-paired dominating graph, Induced-paired dominating set, Induced-paired domination number.
2010 MSC: Primary 05C69; Secondary 05C38, 05C60, 05C76.

1Introduction

Let G be a graph. Wongsriya et al. [3] defined the Γ-total dominating graph of G, and the authors
presented the Γ-total dominating graphs of some paths. Eakawinrujee and Trakultraipruk [1] definedthe Γ-paired dominating graph of G, and they presented the Γ-paired dominating graphs of some paths.
In [2], we introduced the Γ-induced-paired dominating graph of G, and we presented the Γ-induced-paired
dominating graphs of all paths. In this paper, we present the Γ-induced-paired dominating graphs of all
cycles.

2

Main Results

Let Cn = v0 v1 v2 · · · vn−1 v0 be a cycle. First, we give the upper-induced paired domination number of acycle. Since at most two vertices of any three consecutive vertices of a cycle can be in an induced-paired
dominating set, we get the following lemma.
 Lemma 2.1. Let n be a positive integer such that n ≥ 3 and n 6= 5. Then Γip (Cn ) = 2 n3 .
Obviously, IP DΓ (C3 ) is the cycle with three vertices, and IP DΓ (C5 ) is an empty graph. Next, we
determine the Γ-induced-paired dominating graphs of the other cycles.
Theorem 2.2. Let k ≥ 2 be an integer. Then IP DΓ (C3k ) ∼= 3K1 .
For example, the graph IP DΓ (C6 ) is shown in Figure 1.

Figure 1: The Γ-induced-paired dominating graph of C6100

S. Sanguanpong et al.

Short Title

TJCDCGGG2021

Theorem 2.3. Let k be a positive integer. Then IP DΓ (C3k+1 ) ∼= C3k+1 .
For example, the graph IP DΓ (C4 ) is shown in Figure 2.

Figure 2: The Γ-induced-paired dominating graph of C4
Let k be a positive integer. The stairgrid of size k, denoted by Sk , is the subgraph of the Cartesian
(1)
(2)
(3)
product of the paths u1 u2 · · · uk and v1 v2 · · · vk induced by {(ui , vj ) : 1 ≤ i ≤ j ≤ k }. Let Sk , Sk , Sk ,
(4)
(i)
(i)
and Sk be stairgrids of size k, and Cx,y the vertex at position (x, y) of Sk . The spiral-wristband
(1)
(2)
(3)
(4)
of size k, denoted by SWk , is the graph obtained from the stairgrids Sk , Sk , Sk , and Sk by joining
(i)
(i+1)
(1)
(4)
the vertex Cx,k to C1,x−1 for 2 ≤ x ≤ k and 1 ≤ i ≤ 3, and the vertices Cx−1,y−1 and Cx,y are the samefor 2 ≤ x ≤ y ≤ k.
Theorem 2.4. Let k ≥ 2 be a positive integer. Then IP DΓ (C3k+2 ) ∼= SWk−1 .
For example, the graph IP DΓ (C11 ) is shown in Figure 3.

Figure 3: The Γ-induced-paired dominating graph of C11

References[1] P. Eakawinrujee and N. Trakultraipruk,
Γ-paired dominating graphs of some paths,
MATEC web of conferences, 189 (03029), 2018, pp. 1–7. https://doi.org/10.1051/matecconf/201818903029[2] S. Sanguanpong, ”Γ-induced-paired dominating graphs of paths,” M.S. thesis, Dept. of Maths. and Stat.,Thammasat Univ., Pathumthani, Thailand, 2018. https://doi.org/10.14457/TU.the.2018.420[3] A. Wongsriya, W. Kositwattanarerk, and N. Trakultraipruk, Γ-total dominating graphs of some paths,
Proceedings of annual pure and applied mathematics conference 2017 (APAM 2017), Chulalongkorn University, June 28–30, 2017, pp. 266–273.

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The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Biased Domination Game
Tharit Sereekiatdilok1 , and Panupong Vichitkunakorn1
1

Applied Analysis Research Unit, Division of Computational Science, Faculty of Science, Prince of Songkla
University, Songkhla 90110, Thailand
tharit.yoii@gmail.com (Tharit Sereekiatdilok), panupong.v@psu.ac.th (Panupong Vichitkunakorn)

Abstract
We introduce an extended version of a domination game on a graph, called a biased domination
game, in which Dominator and Staller play more than one move in each turn. We show some relationsof biased game domination numbers for different biases.
Keywords: biased domination game, biased game domination number, maximal move, minimal move.
2010 MSC: Primary 91A46; Secondary 05C57, 91A43.

1Introduction

A domination game was introduced in [1] as a game of two players, called Dominator and Staller, ona graph. The players take turns to perform a move by choosing a vertex in the graph. Vertices in theclosed neighborhood of a chosen vertex are said to be dominated. A move u is legal if it creates at least
one new dominated vertex. In other words, its closed neighborhood N [u] is not contained in the closed
neighborhood of vertices set which have been chosen before. That is, for a sequenceof previously pickedSn−1moves u1 , u2 , u3 , . . . , un−1 , the player can pick a move un if and only if N [un ] * i=1 N [ui ]. The gameis ended when all vertices in the graph are dominated. Dominator tries to end the game as soon as
possible, while Staller tries to prolong the game. In the domination game, if Dominator starts the game,
this game is said to be Game 1. Otherwise, it is said to be Game 2. If both of players play optimally ina domination game on a graph G, the number of moves when the game is ended is called the domination
numbers, denoted by γg (G) and γg0 (G) in a Game 1 and Game 2, respectively.
In this work, we introduce a variation of a domination game called a (δ, σ)-biased domination gameor simply called a (δ, σ)-biased game where Dominator and Staller must pick δ and σ moves for each turn(except the last turn of a game), respectively. The (δ, σ)-biased game domination numbers are denoted0by γ(δ,σ) (G) for Game 1 and γ(δ,σ)(G) for Game 2. The main goal of this work is to compare the biased
game domination numbers γ(δ,σ) (G) and γ(δ+1,σ) (G). Similarly, we compare the biased game domination
numbers γ(δ,σ) (G) and γ(δ,σ+1) (G).

2Preliminaries

First, we introduced some definitions and properties in domination games and biased domination game.
Definition 2.1 ( [1]). Let G be a graph and S be a subset of V (G). A partially dominated graph G|S
is a graph G in which all vertices in S are already dominated.
Theorem 2.2 (Continuation Principle [1]). Let G be a graph and A, B ⊆ V (G). If A ⊆ B, thenγg (G|A) ≥ γg (G|B) and γg0 (G|A) ≥ γg0 (G|B).
The proof of Theorem 2.2 also holds for a biased domination game.
Theorem 2.3 (Continuation Principle of the Biased Domination Game). Let G be a graph and A, B ⊆V (G). If A ⊆ B, then γ(δ,σ) (G|A) ≥ γ(δ,σ) (G|B).
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Biased Domination GameTJCDCGGG2021

In 2010, Bresǎr, Klavžar and Rall considered a domination game such that Dominator (resp. Staller)
is allowed, but not obligated, to skip exactly one move in the game. That is, there is at most one turnsuch that Dominator (resp. Staller) may decide to pass. After the game end, the number of moves in agame where both players are playing optimally, is denoted by γgdp (G) (resp. γgsp (G)). We call this gamethe Dominator-pass game (resp. Staller-pass game).
We define the pass game for a biased domination game as follows.
Definition 2.4 ( [1]). In a (δ, σ)-game on a graph G, if Staller is allowed to pass the moves (except thefirst move of each turn) at most n moves, then we define such game as an n-Staller-pass (δ, σ)-game orsp(n)-(δ, σ)-game. The numbers of moves in an sp(n)-(δ, σ)-game when both players play optimally are0
denoted by γsp(n),(δ,σ) (G) in game 1 and γsp(n),(δ,σ)(G) in game 2. Similar notation, dp(n), is used forn-Dominator-pass games.
We define two special types of moves in a biased domination game.
Definition 2.5. We say that a move is minimal if it dominates exactly one new vertex.
Definition 2.6. We say that a move u is maximal if it dominates at least one new vertex w such that
if u0 is another move dominating w then all new vertices dominated by u0 are also dominated by u. In
other words, u is always a better move among all moves dominating w.
If A is the set of all dominated vertices by previous moves, we haveN [u0 ] \ N [A] ⊆ N [u] \ N [A](2.1)

for all legal moves u0 dominating w.

3

Main Results

We get the following results similar to [1].
Theorem 3.1. For any graph G and i ≥ 0, if Staller can always make a minimal move, thenγsp(i+1),(δ,σ) (G) ≤ γsp(i),(δ,σ) (G)andγ(δ,j) (G) ≤ γ(δ,σ) (G)for all j ≤ σ.
Theorem 3.2. For any graph G and i ≥ 0, if Dominator can always make a maximal move (except thefirst move), thenγdp(i+1),(δ,σ) (G) ≥ γdp(i),(δ,σ) (G)andγ(j,σ) (G) ≥ γ(δ,σ) (G)for all j ≤ δ.
These results also hold for Game 2. Moreover, we can applied them to [2] and [3].
Acknowledgments. The authors are grateful to Division of Computational Science, Faculty of Science,Prince of Songkla university and Development and Promotion of Science and Technology Talents Project(DPST) for their support.

References[1] B. Brešar, S. Klavžar, and D.F. Rall, Domination game and an imagination strategy, SIAM Journal on
Discrete Mathematics 24 (2010), no. 3, 979–991.
[2] N. Chantarachada and C. Worawannotai, Domination game on powers of cycles, Thai Journal of Mathematics
2020 (2020), no. Special Issue, 261–268.
[3] Gašper Košmrlj, Domination game on paths and cycles, ARS Mathematica Contemporanea 13 (2017), no. 1,
125–136.

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The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Unidirectional Monotonic Paths through Specified Points
in Labeled Point Sets in Convex Position
Toshinori Sakai1
1

Department of Mathematics, Tokai University, Japansakai@tokai-u.jp

Abstract
Let P be a set of n points in convex position that are labeled with n distinct real values. Then√for any point p ∈ P , there is a unidirectional non-crossing path connecting at least n points of P ,containing p, along which the labels of points monotonically increase (or decrease). In other words,
for any circular permutation with n distinct real-valued terms and for any term k in it, there exists√
a monotone subsequence of length at least n that contains k.
Keywords: labeled point set, convex position, monotonic path.

2
32010 MSC: Primary 52C10; Secondary 11B99.

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Introduction
91

Let P be a set of points in the plane. We denote by conv(P ) the convex hull of P . P is said to be inconvex position if its elements are the vertices of conv(P ). P is called a labeled point set if a number,called a label, is assigned to each point. The label of 5point p will be denoted by 7l(p). All point sets P
considered here are labeled and in convex position, and the labels of the elements of P are all diﬀerent
108real numbers. |P | will be denoted by n.
A polygonal line connecting k points p1 , . . . , pk of P is called 12a monotonic path of length k if thesequence l(p1 ), l(p2 ), . . . , l(pk ) is monotonically increasing or decreasing (Figure 1(a); note that the lengthis not defined as the number of edges, but the number of vertices). Also, a polygonal line is called aunidirectional path if it connects points that appear in the same order along the perimeter of conv(P )(so, it has no self-intersection; Figure 1(b)).

2
3

2
4

6

3
6

11

7
10

8

10

8

9

5

7

5

11

1

9

1

4

12

12

(a)

(b)

Figure 1: (a) A (non-crossing) monotonic path and (b) a unidirectional monotonic path.

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Unidirectional Monotonic Paths

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In 1935, Erdős and Szekeres [2] showed that any sequence of (l − 1)(m − 1) + 1 distinct real numbers
contains either a monotonically increasing subsequence of length at least l, or a monotonically decreasing
subsequence of length at least m (these bounds are tight). From this, it follows that any sequence of ndistinct
real numbers contains a monotonically increasing or decreasing subsequence of length at least√n.
The problem of finding non-crossing monotonic long paths in labeled point sets is a natural variationof the theorem, and this problem was first studied by Czyzowicz et al. [3]. They proved√that any labeledpoint set P in convex position contains a non-crossing
monotonic path of length at least 2n. This bound√
was improved by Sakai and Urrutia√[4] to 3n − 3/4 − 1/2 (they conjecture that there is a non-crossing
monotonic path of length at least 2 n − 1) by giving a simple proof for a result by Chung [1] concerning
the length of a longest unimodal subsequence in a sequence. Sakai and Urrutia [5] also proved that for
any labeled point set P √in convex position and for any element p ∈ P , there is a non-crossing monotonic
path of length at least 2(n − 1) which passes through p.

2

Results

In this talk, we show the following result on unidirectional monotonic paths:
Theorem 2.1. Let P be a set of n labeled points in√ convex position. Then for any point p ∈ P , there isa unidirectional monotonic path of length at least n that passes through p. (This bound is tight.)
This implies that for any circular permutation with n distinct√ real-valued terms and for any term k init, there exists a monotone subsequence of length at least n that contains k.
We also show some results on disjoint non-crossing monotonic paths (not necessarily unidirectional)
for labeled point sets in convex position.

References[1] Chung, F.R.K.: On unimodal subsequences. J. Combin. Theory Ser. A 29 (1980) 267–279[2] Erdős P., Szekeres G.: A combinatorial problem in geometry. Compositio Math. 2 (1935) 463–470[3] Czyzowicz J., Kranakis E., Krizanc D., Urrutia J.: Maximal length common non-intersecting paths. Proc.
Eighth Canadian Conference on Computational Geometry, August 1996, Ottawa, 180–189[4] Sakai T., Urrutia J.: Monotonic polygons and paths of weighted point sets. Computational Geometry, Graphs
and Applications (LNCS 7033, 2011) 164–175[5] Sakai T., Urrutia J.: Long monotonic paths and polygons in weighted point sets. The 16th Japan Conference
on Discrete and Computational Geometry and Graphs, September 2013, Tokyo, 68–69

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The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Logical Matrix Representations in Map Folding
Yiyang Jia1 , Jun Mitani2 , and Ryuhei Uehara3
1

Faculty of Science and Technology, Seikei University, Tokyo 180-8633, Japan
Faculty of Engineering, Information and Systems, University of Tsukuba, Ibaraki 305-8573 Japan3
School of Information Science, JAIST, Ishikawa 923-1292 Japans02966@cc.seikei.ac.jp (Yiyang Jia), mitani@cs.tsukuba.ac.jp (Jun Mitani), uehara@jaist.ac.jp (Ryuhei
Uehara)2

Abstract
Logical matrices are binary matrices often used to represent relations. In the map folding problem,
each folded state corresponds to a unique partial order on the set of squares and thus could be
described with a logical matrix. On the applicational level, such representations provide us an
intuitive recognition of map folding. For instance, using the logical matrices, we can give a precise
mathematical description of a folding process, such that some problems (ex., how to check selfpenetration; how to deduce a folding process according to a given order of the squares) in map
folding can be solved in a mathematical and natural manner. Using this representation, we canestablish the correspondence between folded states and logical matrices on a category level. The
constraint, which makes the correspondence (functor) one-to-one, helps us to embed the category ofpartly folded states to a generally used category, the category of relations Rel. Such analysis would
develop a deep mathematical understanding of map folding.
Keywords: map folding, logical matrix representation, category of relations.
2010 MSC: Primary xxXxx; Secondary xxXxx, xxXxx, xxXxx.

1

Introduction

In this extended abstract, we first introduce a logical matrix representation for map foldings, i.e., foldings
from a rectangular piece of paper embedded with a grid pattern to the shape of a unit square [1]. A
logical matrix is a binary matrix. The set of the logical matrices form a semiring under the addition
and multiplication operations on the semiring (Ω = {0, 1}, +, ·). They are naturally associated withthe category of relations Rel. We also propose a natural transformation from the category of partly
folded states of a map to Rel. As far as we have surveyed, there is so far no research associating
foldings in origami to concrete categories, like the category of relations (Rel) and the category of finite
Hilbert spaces (FHilb). In fact, not only in the field of origami, these two categories, Rel and FHilb
have not been really valued until the contemporary development of quantum computing and its concise
expressions using the category language. This extended abstract gives the first step in the direction of
making correspondence between foldings and important categories in computational science subjects.
There is the potential possibility that according to the category correspondence, the flat-folding of apiece of paper may have something to do with quantum computing.
In contemporary mathematics, the logical matrices are put together with some particular categories.
New discoveries show that both the nondeterministic classical computing, handled within the categoryof relations, Rel, and the quantum computing, handled within the category of Hilbert spaces, Hilb,
have their corresponding matrix representations with similar matrix operations and algebraic structures.
Rel corresponds to the logical matrices while FHilb corresponds to complex matrices [2]. Let us alignthem together: logical matrices vs complex matrices, Rel vs FHilb, and map folding vs flat-folding.
In each pair, the former is a degenerated case of the latter. In this extended abstract, we managed toassociate map folding with logical matrices and Rel. This result makes it natural to use general researchmethods of Rel to study map folding. For example, the logical matrix representation brings some newand simple solutions to some map folding related problems.
Moreover, it is hopeful of associating flat-folding with complex matrices and FHilb in the future.
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LM-Representations in Map Folding

1

2

3
(1-1)

4

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2
3
4
1
(1-2)

Figure 1: (1-1) and (1-2) illustrate a map of size 1 × 4 and one of its possible final state; the right side
is its adjacency representation A and entire representation E

2

Logical Matrix Representation

An overlapping order of a map is a poset where the partial order means for the “up-down adjacent
relation of squares”. The logical matrix recording the adjacent relations is called the adjacency representation. The transitive closure of this partial order involves all the up-and-down relations (notnecessarily adjacent) of all the squares. Correspondingly, the transitive closure through of the adjacency
representation is called the entire representation. For example, a possible final state for the map
in Figure 1(1-1), illustrated in (1-2), correspond to the poset P1 : 1 < 4 < 3 < 2. We denote it by(1, 2, 3, 4, <). The adjacency representation A and the entire representation E of P1 are illustrated on
the right side of Figure 1.

3

Main results

We have two main results. First, as aforementioned, we proposed the logical matrix representation of
map folding and then established a one-to-one correspondence between the category of partly flat-folded
states and the category of logical matrices. It means that the category of partly flat-folded states could
be viewed as a sub-category of Rel. We use simple folds as the morphisms. A simple fold refers to afold on some continuous uppermost or lowermost layers along a single crease, which does not cause anychange on the layers except for the neighborhoods of the crease. The flow of our work is as follows.
Step 1. Define a logical matrix representation to represent every partly flat-folded state and the final
flatly folded state of a map.
Step 2. Give a method to detect the self-penetrations by detecting the numbers of 0s and 1s insubmatrices of a given logical matrix. This step is necessary because we need to clarify which logicalmatrix is a representation of a partly flat-folded state to build the correspondence from the category ofpartly flat-folded states to Rel by an intermediate of the category of logical matrix representations. The
remaining logical matrices are taken as the objects in the category of logical matrix representations.
Step 3. Define the morphisms in the category of partly flat-folded states as simple folds and define themorphisms in the category of logical matrix representations using the addition operation on the logicalmatrix semiring, which is the same as the one on the semiring (Ω = {0, 1}, +, ·).
Step 4. Establish a one-to-one correspondence between the category of partly flat-folded states
and the category of logical matrix representations. The correspondence is then composited with themonomorphic natural transformation from the category of logical matrix representations to Rel. Thus,
finally we know that the category of partly flat-folded states is isomorphic to a sub-category of Rel.
Second, using the logical matrix representations, we can solve the following two problems: 1. A
problem that asks whether a given order of n squares of a map of size 1 × n is foldable by a sequence ofsimple folds or not. We also obtain one legal sequence if it exists. 2. A problem that asks whether there
exist self-penetrations in the given order.

References[1] E. M. Arkin, M. A. Bender, E. D. Demaine, M. L. Demaine, J. S. B. Mitchell, S. Sethia, and S. S. Skiena,
When Can You Fold a Map?, Computational Geometry: Theory and Applications 29 (2002), no. 1, 23–46.
[2] Chris Heunen and Jamie Vicary, Categories for quantum theory: an introduction, Oxford University Press,
USA, 2019.

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The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Feedback game on Eulerian graphs
Naoki Matsumoto1 and Atsuki Nagao2
1

Research Institute for Digital Media and Content, Keio University, Kanagawa 223-8523, Japan
Faculty of Core Research, Ochanomizu University, Tokyo 112-8610, Japannaoki.matsumo10@gmail.com (N. Matsumoto), a-nagao@is.ocha.ac.jp (A. Nagao)
2

Abstract
In this paper, we introduce a two-player impartial game on graphs, called feedback game, which isa variant of generalized geography. The feedback game can be regarded as undirected edge geography
with an additional rule that the first player who goes back to the starting vertex wins the game. We
consider feedback game on an Eulerian graph since the game ends only by going back to the starting
vertex. We first show that it is PSPACE-complete in general to determine the winner of the feedback
game on Eulerian graphs even if its maximum degree is at most 4. In the latter half of the paper,
we discuss the feedback game on two subclasses of Eulerian graphs, i.e., triangular grid graphs andtoroidal grid graphs.
Keywords: Feedback game, Edge geography, Eulerian graph, Triangular grid graph, Toroidal grid graph.
2010 MSC: Primary 05C07; Secondary 05C45, 91A43, 91A46.

1Introduction

All graphs considered in this paper are finite, loopless and undirected unless otherwise mentioned. A
graph G is Eulerian if each vertex of G has even degree. For other basic terminology in graph theory,
we refer to [6].
In combinatorial game theory, impartial games have been well studied for a long time, where a gameis impartial if the allowable moves depend only on the position and not on which of the two players iscurrently moving. So far, many interesting impartial games have been found; e.g., Nim [4], Kayles [8]and Poset game [14]. The most famous result in this area is the Sprague-Grundy theorem [11,15] statingthat every impartial game (under the normal play convention) is equivalent to a one-heap game of Nim.
There are also many interesting games played on graphs as for example; Vertex Nim [7], Ramsey game [9]and Voronoi game [16]. For more details and other topics, we refer the reader to survey several books
and articles [1–3, 5].
One of the most popular impartial games on graphs is generalized geography. Generalized geography
is a two-player game played on a directed graph D whose vertices are words and xy ∈ A(D) if and only
if the end character of a word x is the first one of y, where A(D) is the set of arcs of D. For example, if
x is “Japan” and y is “Netherlands”, then xy ∈ A(D) but yx ∈/ A(D). In this setting, the game begins
from some starting word and both players alternately extend a directed path using unused words. The
first player unable to extend the directed path loses. It is PSPACE-complete to determine the winner ofgeneralized geography [12]. Moreover, several variants of generalized geography have been considered,e.g., planar generalized geography [12], edge geography [13] and undirected geography [10]. It is also
known that for each of above variants is PSPACE-complete to determine which player wins the game
except undirected vertex geography; we can determine the winner in polynomial time.
In this paper, we consider a new impartial game on a graph, called feedback game, which is a variantof undirected edge geography.
Definition 1.1 (Feedback game). There are two players; Alice and Bob, starting with Alice. For a given
connected graph G with a starting vertex s, a token is put on s. They alternately move the token on avertex u to a neighbor v of u and then delete an edge uv. The first player who moves the token back tos or to an isolated vertex (after removing an edge used by the last move) wins the game.
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Feedback game on Eulerian graphs

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In this paper, we investigate feedback game on Eulerian graphs. Note that if a given connected graph
G is Eulerian, then the game does not end until the token goes back to the starting vertex s, and further
observe that Bob always wins feedback game on any connected bipartite Eulerian graph (cf. [10]): Let
G be a connected bipartite Eulerian graph, and so, all vertices of G are properly colored by two colors,
black and white. Without loss of generality, we may suppose that the starting vertex is colored by black.
Throughout the game on G, a token is always moved to a white (resp., black) vertex by Alice (resp.,
Bob). Thus Bob necessarily wins the game.

2

Main results

Here main results are in this paper. Though definitions of technical terms are omitted, one can see themon ArXiV version∗ . The conjecture still unsolved. We hope one enjoy solving this open problem.
Theorem 2.1. It is PSPACE-complete to determine whether there exists a winning strategy for the firstplayer in feedback game, even if the given graph is Eulerian.
Theorem 2.2. If n 6= 2m − 3 with m ≥ 2, then Bob wins the game on the triangular grid graph Tn witha starting vertex v00 .
Lemma 2.3. There is no even kernel graph of Tn that does not include v00 when n > 1.
Theorem 2.4. Alice wins feedback game on T1 and T5 with a starting vertex v00 .
Theorem 2.5. If n = 2m − 3 with m ≥ 2, then the triangular grid graph Tn has no even kernel graph.
Theorem 2.6. If gcd(m, n) = c > 1, then Bob can win feedback game on m × n toroidal grid graph.
Theorem 2.7. If m = 2 or m = 3, and gcd(m, n) = 1, then Alice can win feedback game on m × n
toroidal grid graph.
Theorem 2.8. If gcd(m, n) = 1, then there exists no even kernel graph of m × n toroidal grid graph.
Conjecture 2.9. Alice can win feedback game on m × n toroidal grid graph. if gcd(m, n) = 1.

References[1] B. Allen and M.A. Nowak, Games on graphs, Eur. Math. Soc. 1 (2014), 113–151.
[2] C. Berge, Combinatorial games on a graph, Discrete Math. 151 (1996), 59–65.
[3] E.R. Berlekamp, J.H. Conway and R.K. Guy, Winning Ways for Your Mathematical Plays, A K Peters/CRC
Press, 4 vols. (1982 – 2004).
[4] C.L. Bouton, Nim, a game with a complete mathematical theory, Ann. of Math. 3 (1901), 35–39.
[5] J.H. Conway, On Numbers and Games, A K Peters/CRC Press (2000).
[6] R. Diestel, Graph Theory (fifth edition), Graduate Texts in Mathematics 173, Springer, 2016.
[7] E. Duchêne and G. Renault, Vertex Nim played on graphs, Theoret. Comput. Sci. 516 (2014), 20–27.
[8] H.E. Dudeney, The Canterbury puzzles, Dover, puzzle 73 (2002), 118–119 (Originally published in 1908).
[9] P. Erdös and J.L. Selfridge, On a combinatorial game, J. Combin. Theory Ser. A 14 (1973), 298–301.
[10] A.S. Fraenkel, E.R. Scheinerman and D. Ullman, Undirected edge geography, Theoret. Comput. Sci. 112(1993), 371–381.
[11] P.M. Grundy, Mathematics and games, Eureka 2 (1939), 6–9.
[12] D. Lichtenstein and M. Sipser, GO is Polynomial-Space Hard, J. ACM 27 (1980), 393–401.
[13] T.J. Schaefer, On the complexity of some two-person perfect-information games, J. Comput. System Sci. 16(1978), 185–225.
[14] M. Soltys and C. Wilson, On the complexity of computing winning strategies for finite poset games, Theory
Comput. Syst. 48 (2011), 680–692.
[15] R.P. Sprague, Über mathematische Kampfspiele, Tohoku Math. J. 41 (1935), 438–444.
[16] S. Teramoto, E.D. Demaine and R. Uehara, The Voronoi game on graphs and its complexity, J. Graph
Algorithms Appl. 15 (2011), 485–501.
∗ https://arxiv.org/abs/2002.09570

109

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Solving teleportation mazes with limited visibility
Robert D. Barish1 , and Tetsuo Shibuya1
1

Division of Medical Data Informatics, Human Genome Center, Institute of Medical Science, University ofTokyo, 4-6-1 Shirokanedai, Minato-ku, Tokyo 108-8639, Japanrbarish@ims.u-tokyo.ac.jp (R. Barish), tshibuya@hgc.jp (T. Shibuya)
Abstract

We consider what we denote the Embedded Agent Limited Visibility Teleportion Maze (EALVTM)
problem of efficiently utilizing a player-controlled agent, with distance-limited vision, to discover andsubsequently navigate to a target of interest by moving along the edges of a rectangular or triangularmesh. Here, we prove an N P -hardness result for the optimization version of EALVTM where theobjective is to minimize the total number of edge-wise hops, determine a set of cases where the problem of finding an object on a given mesh becomes Fixed-Parameter Tractable (FPT), and show how
recent advances in low-distortion stochastic embeddings of higher-genus graphs in the plane [3,4] canbe used to extend existing planar graph coverage and path search algorithms to treat the EALVTM
problem on meshes embedded in surfaces having bounded orientable or non-orientable genus.
Keywords: Coverage Path Planning, Graph Coverage, Path Search, A-star, D-star.
2010 MSC: Primary 05C90; Secondary 05C85, 05C57, 05C10.

1

Extended Abstract

We introduce the Embedded Agent Limited Visibility Teleportion Maze (EALVTM) problem of efficiently solving the type of teleportation maze that has been part of the aesthetic of adventure androle-playing games since late 1970’s titles such as William Crowther’s Colossal Cave Adventure and Infocom’s Zork. Here, controlling a single agent with distance-limited vision, our objective is to discoverand subsequently navigate to a target, with the fewest possible number of steps, by walking along theedges of a hole-containing rectangular or triangular mesh embedded in an orientable or non-orientablegenus surface. In this context, we assume either a specified or randomized initial placement of the agent,either no or only partial a priori knowledge concerning the agent’s initial surroundings, and time andspace computational resources at most polynomial in the size of the environment.
For visual intuition, in Figures 1(a-c) we show screenshots of the teleportion-based “player’s maze” inChris Avellone and Black Isle Studio’s Planescape: Torment, and in Figure 1(d) we show a screenshot of
the teleportion maze in Kikiyama’s Yume Nikki. Concerning the Planescape: Torment “player’s maze”,
treating pairs of teleportation portals as handles on a manifold with embedded directed paths of sufficient
length that they cannot be seen through without being traversed, we can determine that the “NamelessOne” protagonist is on a surface of genus at least g ≥ 5 (see the remark in the Figure 1(b) caption
concerning the maze’s true topology). Assuming that an exit portal or item of interest can be placed
arbitrarily, we can observe that EALVTM is at least as hard as a search variant of the Coverage Path
Planning (CPP) [2] or milling problem [1]. Furthermore, we can observe that once an exit portal or objectof interest is at least seen across a barrier – chasms being the vehicle for this in both the Planescape:Torment “player’s maze” and the Yume Nikki teleportation maze (see Figure 1(c,d)) – there remains
the non-trivial task of reaching the objective, which may be no easier than discovering it.
Letting n be the total number of nodes in a mesh for a given instance of EALVTM, defining a “step”
of the agent as a move from one node to another along a mesh edge, and affording a budget of αn total
steps, we prove in part that, unless P = N P , no polynomial time algorithm exists for the EALVTM
subproblem of discovering a randomly placed exit portal or object with an average probability ≥ α2 + This work was supported by JSPS Kakenhi grants {20K21827, 20H05967, 21H04871}, and JST CREST Grant JPMJCR1402JST.

110

R. D. Barish and T. Shibuya

Solving teleportation mazes with limited visibility

TJCDCGGG2021

for  > 0 and ∀α ∈ (0, 2]. In particular, we show that this result holds even in the case of planar
undirected teleportation mazes on both rectangular and triangular grids with holes, fixed or random
initial agent placement, and additionally determine a set of cases under which this search subproblem
becomes Fixed-Parameter Tractable (FPT). In terms of algorithms, we show that recent advances in
low-distortion stochastic embeddings of higher-genus graphs on the plane [3, 4] can be used to adapt
existing plane-based algorithms for the CPP problem to higher genus surfaces. We likewise describe how
the heuristic for the Dynamic A* (D*) algorithm [5] and some of its well-known variants can be adapted
to higher genus surfaces to solve the final navigation component of EALVTM once an exit or object ofinterest is discovered.

Out[ ]=

(a)

(b)

(c)

(d)

Figure 1: Author-taken screenshots of teleportation puzzles from (a-c) Chris Avellone and Black Isle
Studio’s Planescape: Torment, and from (d) Kikiyama’s Yume Nikki : (a) map of the “player’s maze” inPlanescape: Torment with black “fog-of-war” covering unexplored regions, where the player character
positioned at the green dot has just traversed a teleportation portal bridging the pair of gates indicatedby the (author added) white downward arrows (note that portals cannot be seen through without being
traversed); (b) nearly fully explored map of the “player’s maze” (due to teleporter-induced state changes,
the true maze consists of 12 copies of this planar surface embedded closely enough together in Rn thatthe overlapping areas on all planes can be viewed simultaneously by the agent); (c) the “NamelessOne” protagonist (“Adahn”) looking across a chasm at the exit teleportation portal; (d) the Yume Nikki
protagonist Madotsuki in a teleportation maze – where the red tiles serve as portals between disconnected
floating blocks – peering across a gap at a block inhabited by the character “Warpo”.

References[1] E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling.
Comput. Geom., 17(1-2):25–50, 2000.
[2] E. Galceran and M. Carreras. A survey on coverage path planning for robotics. Robot. Auton. Syst.,
61(12):1258–1276, 2013.
[3] Y. Makarychev and A. Sidiropoulos. Planarizing an unknown surface. Proceedings of the 15th international
workshop APPROX and 16th international workshop RANDOM, pages 266–275, 2012.
[4] A. Sidiropoulos. Optimal stochastic planarization. Proceedings of the 51st annual symposium on Foundationsof Computer Science (FOCS), pages 163–170, 2010.
[5] A. Stentz. Optimal and efficient path planning for partially-known environments. Proceedings of the IEEE
International Conference on Robotics and Automation (ICRA), pages 3310–3317, 1994.

111

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Dominect: A Simple yet Deep 2-Player Board Game
Oswin Aichholzer1 , Maarten Löffler2 , Jayson Lynch3 , Zuzana Masárová4 , Joachim Orthaber1 ,Irene Parada5 , Rosna Paul1 , Daniel Perz1 , Birgit Vogtenhuber1 , and Alexandra Weinberger1
1

Graz University of Technology, Austria. 3 University of Waterloo, Canada. 5 TU Eindhoven, the Netherlands.
4
Utrecht University, the Netherlands.
IST Austria, Austria.
{oaich,ropaul,daperz,bvogt,aweinber}@ist.tugraz.at (O.A., R.P., D.P., B.V., A.W.), m.loffler@uu.nl(M.L.), jaysonl@mit.edu (J.L.), zuzana.masarova@ist.ac.at (Z.M.), joachim.orthaber@student.tugraz.at(J.O.), i.m.de.parada.munoz@tue.nl (I.P.)
2

Abstract
In this work we introduce the perfect information 2-player game Dominect, which has recently
been invented by two of the authors. Despite being a game with quite simple rules, Dominect reveals ahigh depth of complexity. We report on first results concerning the development of winning strategies,
as well as a PSPACE-hardness result for deciding whether a given game position is a winning position.
Keywords: Connection game, 2-player game, PSPACE-hardness.
2010 MSC: Primary 91A46; Secondary 91A05.

1Introduction and Game Rules

Dominect is a finite, deterministic perfect information 2-player game developed by Oswin Aichholzer
and Maarten Löffler during the 33rd Bellairs Winter Workshop on Computational Geometry in 2018. It
belongs to the class of connection games, where a typical goal is to form a path connecting two opposite
sides of the game board [5]. Prominent members of this class are Hex (aka Con-tac-tix), TwixT, and Tak.
The Rules of Dominect. Dominect is a two-player game that is played ona rectangular board of m × n squares, where the board size m, n ≥ 3 can vary.
The name Dominect comes from Domino and Connect, where a domino is 2
squares long and 1 square wide and can be placed horizontally or vertically on
empty neighboring squares of the board. The goal is to connect opposite sidesof the board by placing dominoes of the player’s color (red or blue) before theopponent manages to connect the other two board sides by dominoes of theircolor. Dominoes are connected if they are adjacent either vertical, horizontal
or diagonal, that is, touching at a corner is sufficient. Starting with red, thetwo players alternatingly place one domino of their color at a time (from an Figure 1: A connecting
unlimited supply) until one of the players wins or until no more domino can chain for red.
be placed – a draw.
A detailed official book of rules with several illustrations is available on the game’s homepage athttps://dominect.ist.tugraz.at where also some variations of the game can be found.
Dominect is in several ways different from other popular connection games like Hex [3]: For example, adraw is possible for Dominect and also a red connecting chain could cross a blue connecting chain, thatis, both players could simultaneously connect their board sides on the same board. Strategy stealing, anapproach used to show that the second player cannot force a win for many games [1, 2], cannot be appliedfor Dominect. The reason is that each placed domino blocks several other potential domino placements(not only one). So giving one move away by placing a domino somewhere, as used in strategy stealing,
can be a major obstacle and change the whole strategy.

112

2

Winning Strategies and PSPACE-Hardness

It might seem natural to play on quadratic boards (that is, m = n). But it turns out that, at least forsmall sizes, an n × (n − 1) board is more balanced, where the first player has to play the distance n.
Theorem 2.1. For all boards of size m × n with m ≤ 9 or n ≤ 8, except 1 × 1, 3 × 1, 4 × 1, one of theplayers can force a win (assuming that they both play perfectly). Table 1 lists who can force the win.
1 2 3 4 5 6 7 8 9 10 11
Proof ideas. We assume w.l.o.g. that the red player just made a move (andhas not yet won). We want to know for the resulting game position D if the 12
red player can force a win, regardless of what the next moves of the blue player 3
are. If yes, we call D a winning position (for the red player). Let CL = C1 4
5
to CR = Ck be the k ≥ 2 red connected components (sets of red dominoes 6
and board sides which are connected) in D, where CL and CR contain the 7left and right side of the board, respectively. Starting from D, a bridge-move 8
9is a red move that connects two components Ci 6= Cj . Two red moves are 10
independent if they cannot be blocked simultaneously by a single blue domino, 11that is, if no covered square of one move is identical to or sharing an edge Table 1: Depiction of thewith a covered square of the other move.
results of Theorem 2.1.
Hence, if we can find 2` pairwise independent bridge-moves in D and pairthem in a way that after performing either move per pair, the red player has connected CL and CR , andif the blue player cannot connect in at most ` moves, then D is a winning position for the red player. The
results of Theorem 2.1 can be obtained by using this observation in combination with a careful analysis
of different starting moves (see Figure 2 for the 7 × 7 board).
In the full version of this work, we also show that given a game position
D and a set of pairwise independent bridge-moves in D, one can efficiently
compute whether D is a winning position in the above sense. This can also
be used to develop an algorithm that identifies game positions from which one
player can force a win even though D is not a winning position via pairwise
independent bridge-moves. However, while a positive answer of this algorithm
guarantees that a player can force a win, a negative answer does not imply
that the player cannot force a win. We next show that this situation is not
avoidable, as deciding whether or not a player can force a win from a given
game position is computationally hard.
Figure 2: A 7 × 7 board
Theorem 2.2. Deciding whether a given position in the generalized game on
winning strategy.
an n × n board is a winning position is PSPACE-complete.

Proof sketch. To show this, we reduce from the generalized version of Hex,
which is PSPACE-complete [4]. In the reduction each Hex cell corresponds
to a rectangular gadget of dominoes in which exactly one more domino canbe placed (see Figures 3, gadgets marked in black). Placing hexagonal stonesin the Hex grid creates essentially the same connections as placing dominoesof the same colors in the corresponding gadgets.
Acknowledgments.
Research on this topic was initiated at the 33rdthand 34
Bellairs Winter Workshops on Computational Geometry and partially supported by FWF grant W1230 (O.A., I.P., R.P., and A.W.);
NSERC (J.L.); FWF grant Z 342-N31 (Z.M.); and FWF grant I 3340-N35 (I.P., Figure 3: Two rows of
D.P., and B.V.).
three Hex cells each.

References[1] David Blackwell and M. A. Girshick, Theory of games and statistical decisions, John Wiley and Sons, Inc.,New York; Chapman and Hall, Ltd., London, 1954. MR 0070134[2] Cameron Browne, Hex Strategy: Making the Right Connections, 1st ed., CRC Press, 2000.
[3] Ryan B. Hayward and Bjarne Toft, Hex: The full story, 1st ed., CRC Press, 2019.
[4] Stefan Reisch, Hex ist PSPACE-vollständig, Acta Informatica 15 (1981), 167–191.
[5] Leonard Susskind and George Hrabovsky, Connection Games: Variations on a Theme, CRC Press, 2005.

113

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Rainbow and Properly Colored Spanning Trees in
Edge-Colored Bipartite Graphs
Mikio Kano1,∗ and Masao Tsugaki2
1

Ibaraki University, Hitachi, Ibaraki, JapanTokyo University of Science, Tokyo, Japanmikio.kano.math@vc.ibaraki.ac.jp (M. Kano), tsugaki@hotmail.com (M. Tsugaki)
2

We deal with an edge-colored graph G, which have neither loops nor multiple edges. For an edge e ofG, let col(e) denote the color of e. For a vertex v of G, the color degree of v, denoted by degcG (v), is thenumber of colors appeared in the edges incident with v. The minimum color degree of G is defined byδ col (G) = min{degcG (v) : v ∈ V (G)}.
An edge-colored graph G is called rainbow or heterochromatic if all the edges of G have distinct colors.
Moreover, G is called properly colored if no two adjacent edges of G have the same color. Recently thefollowing results were obtained.
Theorem 1 (Cheng, Kano and Wang [2]). Let G be an edge-colored connected graph. Then the following
statements hold.
(i) If δ col (G) ≥ |G|
2 , and if for each color c, the set of edges colored with c forms a subgraph of order|G|
at most 2 + 1, then G has a rainbow spanning tree.
(ii) If δ col (G) ≥ |G|
2 , and if for each color c, the set of edges colored with c forms a star, then G hasa rainbow spanning tree.
(iii) If δ col (G) ≥ |G|
2 , then G has a properly colored spanning tree.
(iv) The condition δ col (G) ≥ |G|
2 in the above statements (i), (ii) and (iii) are sharp.
Notice that the above two statements (ii) and (iii) are easily proved by using the statement (i). In
this paper, we show that the minimum color degree condition given in Theorem 1 can be weakened foredge-colored bipartite graphs as the following theorem.
Theorem 2 ( [4]). Let G be an edge-colored connected bipartite graph. Then the following statements
hold.
(i) Assume that δ col (G) ≥ |G|
3 , for every color c, the set of edges colored with c forms a subgraph of|G|
order at most 3 + 1, and that there are at least |G| − 1 colors appearing in the edges of G. Then G hasa rainbow spanning tree.
(ii) If δ col (G) ≥ |G|
3 + 1, and if for each color c, the set of edges colored with c forms a star, then G
has a rainbow spanning tree.
(iii) If δ col (G) ≥ |G|
3 + 1, then G has a properly colored spanning tree.
col(iv) The condition δ col (G) ≥ |G|(G) ≥ |G|
3 in the above statements (i), and the condition δ
3 + 1 in(ii) and (iii) are sharp. Moreover, the condition that there are at least |G| − 1 colors appearing in theedges of G in (i) is necessary.
New results on rainbow and properly colored spanning trees are found in [3], and many related resultson spanning trees of graphs including the following theorems can be found in the book [1] and a survey [5].
It is known that some sufficient conditions for the usual graph (i.e., a non edge-colored graph) tohave a spanning tree with prescribed property can be weakened for bipartite graphs. In order to explainsome of them, we need some definitions and notation. Let k ≥ 2 be an integer. Then a tree T is called∗ This work was supported by JSPS KAKENHI Grant Number 19K03597.

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M. Kano et al.

Rainbow and Properly Colored Spanning TreesTJCDCGGG2021

an k-ended tree if it has at most k-leaves, and T is called a k-tree if the maximum degree of T is at mostk. Furthermore define{∑}
σk (G) = mindegG (v) : S is an independent set of k vertices of G .
v∈SThen it is known that for an integer k ≥ 2, if a connected graph G1 satisfies σ2 (G1 ) ≥ |G1 | − k + 1,
then G1 has a spanning k-ended tree. On the other hand, it is shown that if a connected bipartite graph
G2 (X, Y ) with |X| ≤ |Y | ≤ |X|+k−1 satisfies σ2 (G2 ) ≥ |G22|−k +1, then G2 has a spanning k-ended tree.
Moreover, it is shown that for an integer k ≥ 2, if a connected graph G3 satisfies σk (G3 ) ≥ |G3 | − 1, thenG3 has a spanning k-tree. On the other hand, it is shown that if a connected bipartite graph G4 (X, Y )with |X| ≤ |Y | ≤ (k − 1)|X| + 1 satisfies σk (G4 ) ≥ |Y |, then G4 has a spanning k-tree. Thus we expectthat Theorem 1 might be weakened for edge-colored bipartite graphs as above, and our theorems showthat this expectation is true.

References[1] J. Akiyama and M. Kano, Factors and Factorizations of Graphs, LNM 2031, Springer (2011)[2] Y. Cheng, M. Kano and G. Wang, Properly colored spanning trees in edge-colored graphs, Discrete Math.,
343, 111629 (2020) .
[3] M. Kano, S. Maezawa, K. Ota, M. Tsugaki and T. Yashima, Color degree sum conditions for properly coloredspanning trees in edge-colored graphs, Discrete Math., 343 112042 (2020).
[4] M. Kano and M. Tsugaki, Rainbow and Properly Colored Spanning Trees in Edge-Colored Bipartite Graphs,
Graphs and Combinatorics, to appear[5] K. Ozeki and T. Yamashita, Spanning trees – A survey, Graphs Combin. 27, 1–26 (2011) .

115

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

The Set Chromatic Number of the
Middle Graph of Extended Stars
Mark Anthony C. Tolentino1 and Gerone Russel J. Eugenio1,2
1

Department of Mathematics, School of Science and Engineering, Ateneo de Manila University, Katipunan
Ave., Loyola Hts., 1108 Quezon City, Philippines
2
Central Luzon State University, Science City of Muñoz, 3119 Nueva Ecija, Philippines
mtolentino@ateneo.edu (M. Tolentino), gerone.eugenio@obf.ateneo.edu (G. Eugenio)
Abstract
Suppose G is a simple, undirected, finite, nontrivial, and connected graph and that c : V (G) → N
of G is a vertex coloring, not necessarily proper, of G. As introduced by Chartrand et al., c is called
a set coloring of G if N C(u) 6= N C(v) for every pair of adjacent vertices u and v; here, N C(x)denotes the set of colors of all the neighbors of the vertex x. Moreover, the set chromatic numberof G, denoted by χs (G), is the fewest number of colors that can be used to construct a set coloringof G. On the other hand, the middle graph of G is defined as the intersection graph of V 0 ∪ E(G),where V 0 is the collection of all singletons each containing a vertex of G. In this paper, we determine
the set chromatic number of the middle graph of extended stars and other families of trees.
Keywords: neighbor-distinguishing coloring, set coloring, middle graph, stars
2010 MSC: Primary 05C15; Secondary 05C05, 05C76.

1Introduction

All the graphs to be considered in this work are simple, undirected, finite, nontrivial, and connected. As
coined in [7], a vertex coloring or edge coloring of a graph is neighbor-distinguishing if it induces a vertex
labelling such that any two adjacent vertices receive distinct labels. Naturally, the classical proper vertex
coloring is neighbor-dsitinguishing.
Among several neighbor-disinguishing colorings that have been introduced (such as those in [2], [3],[5], [6], [7], [8]), the topic of this work is set coloring, which was introduced in [4]. Given a graph G andc : V (G) → N a vertex coloring of G, we have the following definitions:• The neighborhood color set of a vertex v is defined as the set N C(v) = {c(w) : vw ∈ E(G)}.
• The coloring c is called a set coloring of G if N C(v) 6= N C(w) for any two adjacent vertices v andw. If, in addition, |c(V (G))| = k (i.e. c uses k colors), then we call c a set k-coloring of G.
• The set chromatic number of G is denoted by χs (G) and is defined as the smallest integer k forwhich G has a set k-coloring.
Set colorings have been studied in relation to different graph operations involving different graphfamilies. For instance, there have been studies in relation to the join [10, 16], corona and vertex/edge
deletions [4], and comb product [10]. Meanwhile, similar studies have been carried out involving otherneighbor-distinguishing colorings such as the ones in [1], [11], [12], [14], [15], and [17].
Previously, in [9], the authors studied set colorings in relation to a graph operation called middle
graph. Introduced by Hamada and Yoshimura [13], the middle graph M (G) of a graph G is defined tobe the intersection graph of V 0 ∪ E(G), where V 0 is the set of all singletons each containing a vertex of
G. Alternatively, we may think of M (G) as the graph whose vertex set is V (G) ∪ E(G) and in which
two vertices u and v are adjacent if and only if• u and v are adjacent edges in G; or• u ∈ V (G), v ∈ E(G), and u is incident to v in G.
The results in [9] include bounds for χs (M (G)) as well as the set chromatic number of the middle graphof stars, paths, cycles, and tadpoles. This work aims to continue [9] by considering the middle graph ofother families of trees.
116M. Tolentino et al.

2

The Set Chromatic Number of the Middle Graph of Extended StarsTJCDCGGG2021

Main Results

Let m, n be positive integers. We define the extended star graph Tm,n to be the graph obtained byconnecting n pendant vertices to each pendant vertex of the star graph K1,m . Such graphs have been
used, for example, in [18]. We now present one of the main problems of our work:
Problem 2.1. Construct optimal set colorings and determine the set chromatic number of the middle
graph of extended star graphs.
We introduce the following: Let m, n, D, ` be positive integers where m ≥ n + 1 ≥ 2. We denote by
qm,n (D, `), orP
q(D, `) if there is no ambiguity on the values of m and n, the smallest positive integer krthat satisfies i=0 C(k − `, i) ≥ m − ` + D, where r := min{k − `, n + 1}. Our main result on the middle
graph of extended stars is as follows:
Theorem 2.2. Let Tm,n be an extended star graph. Thenn + 1,χs (M (Tm,n )) =min{max{n + 2, q1,2 }, max{n + 1, q3,3 }},n ≥ m ≥ 4,m ≥ n + 1 ≥ 2.

Aside from extended stars, our work also includes the determination of the set chromatic number ofthe middle graph of other families of trees such as banana graphs and some caterpillars.

References[1] J. Aruldoss and S. Mary, Harmonious coloring of middle and central graph of some special graphs, Int. J.
Math. Appl. 4 (2017), 187–191.
[2] S. Arumugam, K. Premalatha, M. Bača, and Semaničová-Feňovčı́ková, Local antimagic vertex coloring of agraph, Graphs Combin. 33 (2017), 275–285.
[3] G. Chartrand, S. English, and P. Zhang, Kaleidoscopic colorings of graphs, Discuss. Math. Graph Theory
37 (2017), no. 3.
[4] G. Chartrand, F. Okamoto, C. W. Rasmussen, and P. Zhang, The set chromatic number of a graph, Discuss.
Math. Graph Theory 29 (2009), no. 3, 545–561.
[5] G. Chartrand, F. Okamoto, and P. Zhang, The metric chromatic number of a graph, Australas. J. Combin.
44 (2009), 273–286.
[6], Neighbor-distinguishing vertex colorings of graphs, J. Combin. Math. Combin. Comput. 74 (2010).

[7], The sigma chromatic number of a graph, Graphs Combin. 26 (2010), 755–773.

[8] G. Chartrand and P. Zhang, Chromatic graph theory, Boca Raton: Chapman & Hall/CRC, 2009.
[9] G. R. J. Eugenio, M. P. Ruiz, and M. A. C. Tolentino, The set chromatic numbers of the middle graph ofgraphs, J. Phys. Conf. Ser., vol. 1836, IOP Publishing, 2021.
[10] B. C. Felipe, A. D. Garciano, and M. A. C. Tolentino, On the set chromatic number of the join and comb
product of graphs, J. Phys. Conf. Ser., vol. 1538, IOP Publishing, 2019.
[11] A. D. Garciano, R. M. Marcelo, M. P. Ruiz, and M. A. C. Tolentino, Sigma coloring and edge deletion, J.
Inf. Process. 28 (2020).
[12] L. Guangrong and Z. Limin, Total chromatic number of one kind of join graphs, Discrete Math. 306 (2006),
1895–1905.
[13] T. Hamada and I. Yoshimura, Traversability and connectivity of the middle graph of a graph, Discrete
Mathematics 14 (1976), 247–255.
[14] F. Hanna, Equitable coloring of graph products, Opuscula Math. 26 (2006), no. 1.
[15] A. Noga and M. Bojan, The chromatic number of graph powers, Combin. Probab. Comput. 11 (1993), 1–10.
[16] F. Okamoto, C. W. Rasmussen, and P. Zhang, Set vertex colorings and joins of graphs, Czechoslov. Math.
J. 59 (2009), no. 4, 929–941.
[17] J. Vivin, M. Venkatachalam, and N. Mohanapriya, On b-chromatic number of some line, middle and total
graph families, International J.Math. Combin. 1 (2016), 116–125.
[18] Saad Yalouz and Vincent Pouthier, Continuous-time quantum walk on an extended star graph: Disorderenhanced trapping process, Physical Review E 101 (2020).

117Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

On rainbow mean colorings of brooms and double brooms
Agnes D. Garciano, Reginaldo M. Marcelo, Mari-Jo P. Ruiz, and Mark Anthony C. Tolentino
Ateneo de Manila University, Quezon City, Philippines{agarciano, rmarcelo, mruiz, mtolentino}@ateneo.edu

Abstract
Let G be a simple connected graph and c an edge coloring that uses positive integers. Forv ∈ V (G), let cm(v) be the average of the colors of the edges incident to v. If cm(v) is an integer
for each v ∈ V (G) and cm(v) 6= cm(u) for any u 6= v ∈ V (G), then c is called a rainbow mean
coloring. In such a coloring c, let rm(c) be the maximum value of cm(v) for v ∈ V (G). The
rainbow mean index of G, denoted by rm(G), is the minimum value of rm(c) among all rainbow
mean colorings c of G. A rainbow mean coloring c of a graph G is called optimal if rm(c) = rm(G),and G is said to be Type 1 if rm(G) = |V (G)|. In this paper, we determine the rainbow mean index
of families of trees that are called brooms and double brooms. We show that aside from P4 and thestars with even order, all other brooms are Type 1. We also show that all double brooms that have
handles longer than 3 are Type 1. We also determine an optimal rainbow mean coloring for suchbrooms and double brooms.
Keywords: rainbow mean coloring, rainbow mean index, brooms, double brooms.
2010 MSC: 05C15, 05C78

1

Introduction and Preliminaries

Let G be a simple connected graph and c an edge coloring that uses positive integers. For each v ∈ V (G),the chromatic mean of v denoted by cmc (v), or simply cm(v), is the average of the colors of the edges
incident to v. If cm(v) is an integer for each v ∈ V (G) and distinct vertices have distinct chromaticmeans, then c is called a rainbow mean coloring. In such a coloring c, the maximum chromatic meanof a vertex is called the rainbow mean index of c and is denoted by rm(c). On the other hand, therainbow mean index of G, denoted by rm(G), is the minimum value of rm(c) among all rainbow mean
colorings c of G.
The rainbow mean coloring of a graph is an example of a vertex-distinguishing coloring which hasreceived increased attention in the last few decades [2]. Chartrand et.al. [1] introduced the rainbow mean
coloring in 2019. They determined the rainbow mean index of paths, cycles, complete graphs, and stars.
Meanwhile, Hallas et.al. [3] determined the rainbow mean index of several bipartite graphs including
prisms, hypercubes, and complete bipartite graphs.
Let n ≥ 3 and k be positive integers, n > k, Pn−k a path of order n − k with terminal vertex u
and K1,k the star with central vertex v. Then the graph of order n obtained from Pn−k and K1,k by
identifying the vertices u and v is called a broom and is denoted by B (n, k). We call the path Pn−k
(as a subgraph of B (n, k)) the handle of the broom, with initial vertex the tip of the broom. If k = 1
then the broom is a path, while if n − k = 1 or 2, it is a star.
Now, let n, k1 , and k2 be positive integers where n ≥ k1 + k2 + 2, k1 ≥ 2, k2 ≥ 2. Let Pn−k1 −k2 be apath with order n−k1 −k2 and terminal vertices u and v. If we identify u with the central vertex of K1,k1
and v with the central vertex of K1,k2 , then the graph of order n obtained is called a double broom
and is denoted by DB (n, k1 , k2 ). We will also call the path Pn−k1 −k2 (as a subgraph) the handle of the
double broom.
For convenience, we call a rainbow mean coloring c of a graph G optimal if rm(c) = rm(G) . Following
the terminology in [3], we say G is Type 1 if rm(G) = n where n is the order of G.
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On rainbow mean colorings . . .

(a)

(b)

TJCDCGGG2021

(c)

(d)
Figure 1: The double room DB(12, 4, 3) with an optimal rainbow mean coloring.

2

Main Results

Lemma 2.1. Let k any positive integer and G a connected graph with rainbow mean coloring c. Let c0
be an edge coloring of G defined by c0 (e) = c (e) + k, for each edge e ∈ E(G). Then c0 is a rainbow mean
coloring of G. Moreover, rm(c0 ) = rm(c) + k.
Lemma 2.2. Let G1 and G2 be connected graphs of order n1 and n2 , rm(G1 ) = r1 , rm(G2 ) = r2 , andwith optimal rainbow mean colorings c1 and c2 , respectively. Suppose ui ∈ V (Gi ), for i = 1, 2, withcm1 (u1 ) = r1 and cm2 (u2 ) = 1. If G is the graph obtained from G1 and G2 by identifying vertices u1
and u2 , then rm(G) ≤ r1 + r2 − 1.
Theorem 2.3. For any positive integers n ≥ 3 and k with 1 ≤ k ≤ n − 1, rm(B(n, k)) = n except forthe cases that B(n, k) = P4 and B(n, k) = K1,n−1 where n is even.
Theorem 2.4. Let n, k1 , and k2 be positive integers where k1 ≥ 2, k2 ≥ 2, and n − k1 − k2 ≥ 5. Thenrm(DB (n, k1 , k2 )) = n.
We illustrate the proof of Theorem 2.4 on DB (12, 4, 3).
1. By Theorem 2.3, rm(B (7, 4)) = 7 and rm(B (6, 3)) = 6.
2. In the proof of Theorem 2.3, it can be shown that B (7, 4) has an optimal rainbow mean coloringc1 where the tip (u1 ) has chromatic mean 7 (Figure 1a).
3. It can also be shown that the broom B (6, 3) has an optimal rainbow mean coloring c2 where thetip (u2 ) has chromatic mean 1 (Figure 1b).
4. We can then apply Lemma 2.2 on the brooms B (7, 4) and B (6, 3) together with the colorings
c1 and c2 , to get the double broom DB (12, 4, 3) by identifying vertices u1 and u2 . By Lemma
2.2, rm(DB (12, 4, 3)) ≤ 7 + 6 − 1 = 12. Since rm(DB (12, 4, 3)) ≥ 12, it now follows thatrm(DB (12, 4, 3)) = 12. Note that an optimal rainbow mean coloring for the double broom is
obtained by first increasing the labels of c2 for B (6, 3) by 6 (Figure 1c) before identifying u1 andu2 .

References[1] G. Chartrand, J. Hallas, E. Salehi, & P. Zhang, Rainbow mean colorings of graphs, Discrete Math. Lett., 2(2019), 18–25.
[2] H. Escuadro, F. Okamoto, P. Zhang, & S. Arumugam, Vertex-Distinguishing Colorings of Graphs: A Surveyof Recent Developments, AKCE Int. J. Graphs Comb., 4:3 (2020), 277–299.
[3] J. Hallas, E. Salehi, & P. Zhang, Rainbow mean colorings of bipartite graphs, Bull. Inst. Combin. Appl., 88(2020), 78–97.

119

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Fractional Graph Capacity
Ghurumuruhan Ganesan11Institute of Mathematical Sciences, HBNI, Chennai, India
gganesan82@gmail.com

Abstract
In this paper we study a generalization of graph capacity which we call as the fractional graphcapacity and use the probabilistic method to determine bounds on the fractional capacity for arbitrary
graphs.
Keywords: Fractional Graph Capacity, Bounds, Probabilistic Method.
2010 MSC: Primary 05C90; Secondary 05C50.

1Introduction

The capacity of a graph was introduced by Shannon to determine the limits of error-free communication
across channels with an underlying confusion graph. Lovász [4] obtained the expression for capacity of
the cycle on 5 vertices using “umbrella” projection methods and since then, many bounds as well as
variants of the capacity have been studied (see, for example, [5] and [1]). In this paper, we introduce thenotion of fractional capacity of a graph and use the probabilistic method to obtain bounds in terms ofthe graph degree parameters.

2

Fractional Graph Capacity

Let G = (V, E) be any connected graph containing #V = n ≥ 3 vertices and for a vertex u, let NG [u]be the set of all neighbours of u, including u. The graph distance between any two nodes u and v isthe number of edges in the shortest path between u and v. A set F ⊂ V of vertices is said to bestable if no two vertices in F are adjacent to each other in G. We denote α(G) to be the independence
number, i.e. the maximum size of a stable set in G. For integer r ≥ 1 let G(r) be the rth strong graphproduct of G obtained as follows: The graph G(r) has vertex set V r and two vertices u = (u1 , . . . , ur )and v = (v1 , . . . , vr ) are adjacent if and only if ui ∈ NG [vi ] for each 1 ≤ i ≤ r. For an integer 1 ≤ k ≤ r,
we now define the subgraph G(r, k) ⊆ G(r) as follows. Two vertices v, u ∈ V r are
Pradjacent in G(r, k)
if and only if u and v are adjacent in G(r) and differ in at most k entries i.e., i=1 1(u1 i 6= vi ) ≤ k,where 1(.)
1 refers to the indicator function. We have the following definition.
Definition 2.1. For a real number 0 < γ ≤ 1 we define the γ−fractional capacity of G to be
1

Θγ (G) := sup (α(G(r, γr))) r .

(2.1)

r≥ γ1

For γ = 1, the term Θ1 (G) =: Θ(G) is the graph capacity as defined in Shannon (1956). For
differentiation, we refer to Θ(G) as the full graph capacity and Θγ (G) as the fractional graph capacity.
In the context of codes, each vertex of G(r) is a codeword of length r and the term γ represents the
maximum fraction of symbols that undergo corruption when passed through a channel with confusiongraph G. The quantity (Θγ (G))r is then the maximum size of a code from G(r) that allows for error freecommunication. For 0 ≤ x ≤ 1 we let H(x) := −x · log x − (1 − x) · log(1 − x) be the entropy function,
where logarithms are to the base two throughout and have the following bounds on the fractional graphcapacity.
120

G.Ganesan.

TJCDCGGG2021

Theorem 2.2. Let G be a connected graph on n vertices and let dav and ∆ be the average and maximum
vertex degree of G, respectively. For any 0 < γ ≤ 1 we have that

γ
Θ(G)
n · max(f (γ, dav ), f (γ, ∆), f (γ, n − 1)) ≤ Θγ (G) ≤ n ·
(2.2)
n
(
−1
x
2H(γ) · xγ
for 0 < γ < x+1
where f (γ, x) :=x−1(x + 1)for x+1 ≤ γ ≤ 1.
Setting γ = 1 in the lower bound of (2.2), we get that Θ(G) ≥ davn+1 which is also obtained via theTurán’s bound. For example
the graph G = C5 , the cycle on 5 vertices, has dav = ∆ = 2 and it is√55≤ Θ 21 (C5 ) ≤ √well-known that Θ(G) = 5 [4]. Therefore setting γ = 21 , we get from (2.2) that 2√
4 .
2
5
We prove the lower bound in (2.2) using the probabilistic method and a maximal stable set argument(the Gilbert-Varshamov argument [3]) and prove the upper bound in (2.2) using a recursion estimate
similar to the Singleton argument [3].
Proof of the lower bound in (2.2): For a uniformly random vector v = (v1 , . . . , vr ) ∈ V r , let Bγ (v) be theset of all vertices adjacent to v in the graph G(r, γr). The vertices {vj }1≤j≤n are mutually independentand uniformly distributed
in V and so the expected degree of vj is dav . Consequently, the expected sizePγrof Bγ (v) is E#Bγ (v) = k=0 kr dkav .
For 0 < γ < 1− dav1+1 , we use the Hamming ball estimate (Proposition 3.3.1, [2] to get that E#Bγ (v) =Pγr r  klog davθrwhere θ = H(γ)+γ
satisfies 0 < θ < 1. For 0 <  = (r) < 1 to be dek=0 k dav ≤ n
log n
termined later, we let A() := {u ∈ V r : #Bγ (u) ≤ nr(θ+) } and obtain from Markov inequalitythat #A() ≥ nr (1 − n−r ) . Let D := {w1 , . . . , wM } ⊆ A() be a stable set of maximum size in G(r, γr).
SM
By the maximality, we must have that the union i=1 Bγ (wi ) = A() and soMXnr 1 − n−r ≤ #A() ≤#Bγ (wi ) ≤ M · nr(θ+) .
i=1

Thus M ≥ n(1−θ−)r (1 − n−r ) and choosing  = √1r , taking rth roots and allowing r → ∞, we getthat Θγ (G) ≥ nf (γ, dav ).

Pr
r k
r
For 1 − dav1+1 ≤ γ ≤ 1, we use the inequality E#Bγ (v) ≤k=0 k dav = (dav + 1) and the
maximality argument as before, to get that Θγ (G) ≥ nf (γ, dav ). To see that Θγ (G) ≥ nf (γ, ∆), we usera similar argument
 r along with the estimate that for any deterministic vector u ∈ V the ballPγr as rabove
size #Bγ (u) ≤ k=0 k ∆ . Finally, a similar argument also shows that Θγ (G) ≥ nf (γ, L − 1) and thiscompletes the proof of the lower bound in (2.2).
Proof of the upper bound in (2.2): Let d = γr and let C ⊂ G(n) be a stable set of maximum
size in G(r, d) and set A(r, d) := #C. We first derive a recursive estimate involving A(r, d). For avertex v ∈ G let C(v) be the set of all vertices in C whose last entry is v. By the pigeonhole principle,
there exists a v0 ∈ G such that #C(v0 ) ≥ A(r,d)
n . We remove the last entry in each vertex of C(v0 ) andcall the resulting set as D(v0 ) ⊂ G(r − 1). The set D(v0 ) is also stable in G(r − 1, d) and this implies
that A(r, d) ≤ n · A(r − 1, d). Continuing iteratively we get A(r, d) ≤ nr−d · A(d, d). Taking rth roots1
and using the fact that supr≥ γ1 (A(d, d)) r = supd≥1 (A(d, d))γd = (Θ(G))γ , we get the upper boundin (2.2).
Acknowledgments. I thank IMSc faculty for crucial comments and also thank IMSc for my fellowships.

References[1] N. Alon and E. Lubetzky, The Shannon capacity of a graph and the independence numbers of its powers,
IEEE Transactions on Information Theory, 52 (2006), 2172–2176.
[2] V. Guruswami,
A. Rudra and M. Sudan,
Essential Coding Theory.
https://cse.buffalo.edu/faculty/atri/courses/coding-theory/book/web-coding-book.pdf, 2019.

Link:[3] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003.
[4] L. Lovász, On the Shannon Capacity of a Graph, IEEE Transactions on Information Theory, 25 (1979),
1–7.
[5] K. Marton, On the Shannon capacity of Probabilistic Graphs, Journal of Combinatorial Theory, Series B,
57 (1993), 183–195.

121

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Unit Tilt Row Relocation in a SquareDavid Caballero1 , Angel A. Cantu2 , Timothy Gomez1 , Austin Luchsinger3 , Robert Schweller1 , andTim Wylie1
1

Department of Computer Science, University of Texas Rio Grande Valley, Edinburg, TX USA
Southwest Research Institute, San Antonio, TX USA
3
Electrical and Computer Engineering Department, University of Texas Austin, Austin, TX USA
2

Abstract
Relocation in the square is the most basic open question related to the tilt model with unit moves
under global uniform movement. It is open even with limited directions. We prove that deciding if apiece can move to the bottom row with only two directions is in P and gives the minimum column.
Keywords: tilt model, robot motion planning, relocation, uniform control, external forces.
2010 MSC: Primary 52C99; Secondary 68R05, 52B20, 68W01.

1Introduction

Tilt assembly with uniform global instructions is currently a popular area of research since individuallycontrolling nanobots may be infeasible or impractical at certain scales. One of the most basic open
questions in the single step tilt model with uniform global movements is whether a tile can be relocatedto specific location on the board (Figure 1a) even without any blocking geometry. We solve a simpler
version of this problem asking whether, with only two directions, a tile can reach the bottom row.
Given a tilt system with an n × n board B, a set of tiles T = {t1 , . . . tm } where each tl = (i, j) s.t.
1 ≤ l ≤ m, 1 ≤ i ≤ n, and 1 ≤ j ≤ n. Given ti , tj ∈ T, ti 6= tj if i 6= j. For shorthand, we use ti,j fortl = (i, j). For just the row or column, we may use tlr and tlc . See [1–3] for model and related work.
Definition 1.1 (First Row Relocation). Given a specific tile to relocate tR at location (r, c) = (tRr , tRc ),the first row relocation problem asks whether a series of transformations can translate tR s.t. tRr = 1.

2

High Level Overview

In order for tile tR to reach the bottom row, there must not be any other tiles beneath it. We first findthe empty column needed, and then, through a process termed knitting, we attempt to move the empty
column beneath tR . We also label the board with sections (Figure 1a). See Figure 1b for examples ofthe empty column. We then label the rows with counts of tiles that may affect relocation (Figure 1c).
Definition 2.1 (Empty Column Ec ). Given a tilt system board configuration S = (B, T) and tiletR = (r, c) where tR ∈ T, define Ec = min{k : c ≤ k ≤ n + 1 s.t. |{ti,j : ti,j ∈ T, i < r, j = k}| = 0}. If
all columns in the BR section have tiles, Ec = n + 1 and enters the board after a west, hW i, movement.
Definition 2.2 (Knitting). The row between the BL and TL section is the knitting row. Knitting is theact of performing hW i movements when every position of the knitting area is occupied by a tile. Thus,tR maintains its position.
Definition 2.3 (Counts). Given a tilt system board configuration S = (B, T), a tile tR ∈ T with location(r, c), and the target empty column Ec . Define the count of a row k as follows.

if k > r (rows above tR )|{ti,j : ti,j ∈ T, i = k, 1 ≤ j < c}|,Count(k, tr,c ) = |{ti,j : ti,j ∈ T, i = k, 1 ≤ j ≤ c}|,if k = r (tR row),|{ti,j : ti,j ∈ T, i = k, 1 ≤ j < Ec }|, if k < r (rows below tR )This basic framework leads to two important lemmas (2.5 2.6). The algorithm will ensure that oneof these is eventually met. The basic idea is Algorithm 1.
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Row Relocation in a Square under Unit Uniform Movement

Top
Left(TL)Top
Right(TR)

Knitting
Row
BottomLeft(BL)

Tile torelocate
BottomRight(BR)
Target Location(a) Board Sections

1
2
1
2
5
3
5
2

(b) Empty column examples
1
2
1
2
5
2
2
5

1
1
2
2
4
1
2
1

(c) Example counts

1
3
0
0
0
1
1
3

2
4
2
3
5
3
4
2

(e) Knitting solution

1
4
0
0
2
3
1
2

(f) Tiles in H(d) Has no solution
1
2
1
2
5
3
4
2

1
2
2
2
3
0
0
0

2
4
2
3
5
3
5
2

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1
3
0
0
2
3
1
2

1
2
0
0
2
23
1
2

(g) Reducing tiles

Figure 1: (a) The board is an n × n square divided into 6 sections based on the tile to relocate. The
four large sections are the Top Left (TL), Top Right (TR), Bottom Left (BL), and Bottom Right (BR)sections. There is also the knitting area (outlined in blue) and the tile to relocate (red dot). When
we move to general relocation, there will also be a target spot (red square). (f) Tiles that enter the
BR section after D movements are included in NR . If the empty column is occupied by a tile after
D movement (i.e., it becomes non-empty), then every tile between the empty column and the next
empty column (shown in red) are also added to NR . Reducing the number of tiles in H is done by hW i
movement. (g) Tiles in H that lead towards another tile beneath it are removed from H.

2.1

Changing the Knitting Row

If neither condition in the lemmas is satisfied, we cannot knit with the tiles in the knitting row. Define
the candidate row as the closest row in the TL section with a count higher than the knitting row.
Definition 2.4 (Candidate Row). Given a tilt system board configuration S = (B, T) and a tile tR ∈ T
at location (r, c). The knitting row may contain up to c − 1 tiles. The closest row (fewest south, hSi,
movements) in the TL section with a count higher than the knitting row is the candidate row.
Result: First Row Relocation
while Lemmas 2.5, 2.6 are unsatisfied do
Determine Ec and counts for each row;
if Lemma 2.5 is satisfied then accept;
if Lemma 2.6 is satisfied then reject;
Set candidate row CR;
Move CR to be the new knitting row;
Algorithm 1: High Level Idea

Lemma 2.5. [Existence] If the count of the knitting area is greater than the counts of the lower
section, then first row relocation is possible.
Lemma 2.6. [Nonexistence] If there exist a countin the bottom sections that is larger than every
count in the TL section and knitting area, then firstrow relocation is impossible.

If we make any movements, we may introduce new tiles that must be considered. We look at new
tiles that may enter the BL section (NL ) and tiles that may enter in the BR section (NR ).
We include the set of tiles in TR that are at most D distance (with hSi moves) from the BR that
increase the counts of the bottom rows (Figure 1f). The BR and TR sections may change if tR moves.
If Ec has tiles after D movements, then we change Ec to the correct column. All tiles that enter the last
row after D movements to the left of the new empty column are included in NR . Let H be the set of
all tiles in either the TL or BL section that are within D hSi movements from the first row. With hW imoves, we may be able to reduce H (Figure 1g). When first row relocation is possible, the sequence
is W u S v W x S y and u, v, x, y must be determined, but are bounded by n. There are ≤ c − 2 possible
candidate rows, and at most n − r moves to bring any CR to the knitting row.
Theorem 2.7. First row relocation of tile tr,c ∈ T on an n × n board can be solved in O(cn + c|T|) time.
Funding. This research was supported in part by National Science Foundation Grant CCF-1817602.

References[1] David Caballero, Angel A. Cantu, Timothy Gomez, Austin Luchsinger, Robert Schweller, and Tim Wylie,
Building patterned shapes in robot swarms with uniform control signals, Proceedings of the 32nd Canadian
Conference on Computational Geometry, CCCG’20, 2020, pp. 59–62.
[2], Hardness of reconfiguring robot swarms with uniform external control in limited directions, Journalof Information Processing (2020).

[3], Relocating units in robot swarms with uniform control signals is pspace-complete, Proceedings of the
32nd Canadian Conference on Computational Geometry, CCCG’20, 2020, pp. 49–55.

123

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Discrete Quantum Systems via Tight Group Frames
and their Geometrization
Francis D. Delloro1 and Job A. Nable2
1

Department of Mathematics, Ateneo de Naga University, Naga City 4400, Philippines
Department of Mathematics, Ateneo de Manila University, Quezon City 1108, Philippines
fdelloro@gbox.adnu.edu.ph (F. Delloro), jnable@ateneo.edu (J. Nable)
2

Abstract
This work uses finite tight group frames to implement a Groenewold-Moyal or star-product formalism of quantum mechanics on finite-dimensional Hilbert space. Tight frames originated in signal
analysis via wavelets and Gabor systems and in quantum mechanics via the discretization of coherent states. Geometrization of the star-product by contravariant tensors is deduced from the Wignertransform of quantum operators arising from frame quantization.
Keywords: Tight frames, discrete quantum mechanics, star-product, phase space quantum mechanics, geometrization.
2010 MSC: Primary 42C15, 81R05; Secondary 81S30, 81R60.

1Introduction

Let H be a Hilbert space. A frame in H is a family
P {ψx : x ∈ X} ⊂ H parametrized by a discrete space X
satisfying the following condition: A||φ||2 ≤ x∈X |hφ, ψx i|2 ≤ B||φ||2 , for some constants 0 < A ≤ B.
If A = B, the frame is
Pcalled a tight frame. In this latter case, a sort of Fourier expansion is provided bytight frames: ϕ = A1 x∈X hψx , ϕiψx . The difference with bases is that frames do not necessarily consistof linearly independent vectors. It is to this aspect that frames owe its utility in many areas of scienceand engineering [2, 4].
This paper has two objectives, to use frames in the star-product formalism of discrete quantum
systems and to propose a way of geometrizing the main objects of the formalism, given by the starproduct, the Lie bracket and the Jordan bracket. It is notable that geometrization of quantum systems
is a long standing program of mathematical physics, one of whose aim is the unification of quantum
mechanics and gravity.

2

Preliminaries

2.1

Phase Space Representation of Quantum Mechanics

The dynamics in classical mechanics is given by Hamilton’s Equation df
dt = {f, H} in terms of the Poissonbracket on the space of functions on M , and the energy H of the system. On the other hand, quantumbb A],
b in terms of the energy H
b of the system anddynamics is given by Heisenberg’s equation ddtA = ~i [H,commutator [·, ·]. The similarity of the dynamics suggests that a basic task of quantum mechanics is tobf from classical observables to self-adjoint operators. However,find a quantization mapping Q : f 7→ A
the works of H. Weyl, E. Wigner, H. Groenewold, E. Moyal and others suggest another point of view,that of quantum mechanics in phase space [5]. The fact that there is a one-to-one correspondence
between the Weyl transform of phase space functions and the Wigner transform of Hilbert-Schmidt
operators allows for theR representation of quantum operators by functions on classical phase space,∞b + y/2idy. To the composition Ab◦Bb corresponds agiven by WAb(q, p) = h1 −∞ e−ipy/~ hq − y/2|A|q
noncommutative and associative product on phase space, making it possible to implement an equivalentquantum theory on phase space, which is most suitable for the analysis of the classical to quantum
correspondence. Moreover, a remarkable parallel with the methods of time-frequency representation of
signal analysis shows the great utility of these ideas.
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2.2

Discrete Quantum Systems via Tight Group Frames and their Geometrization

TJCDCGGG2021

Coherent States and Frames

Coherent states are quantum states nearest in behavior to classical states via minimization of the
Heisenberg Uncertainty Principle. The most basic, but also the most important, coherent states are†
Tb(z)ψ0 = ezba−zba ψ0 , z ∈ C, associated to the quantum harmonic oscillator, introduced by E. Schrödinger
in 1926. Coherent state quantization and frame quantization are the first applications of these objects,
due to F. Berezin and to J. von Neumann.
Nonorthogonal expansions of square-integrable functions on the plane may be obtained by choosing an appropriate discrete subset Ω of the plane and forming the set of vectors Tb(qm , pn )ψ0 , wherezmn P= qm + ipn and ψ0 is a Gaussian function. This set is nonorthogonal but has the property
f = m,n∈Ω hf, Tb(qm , pn )ψ0 iTb(qm , pn )ψ0 , for square integrable function f on the plane (called Gaborexpansion). This collection of vectors is called a frame in Hilbert space. The consideration of framesin finite-dimensional Hilbert spaces is mainly due to numerical implementation of the theory in various
applications, such as quantum optics, quantum information, and signal analysis and data processing [2,4].

3

Main Results

An important class of discrete and finite tight frames are those that arise from unitary group actions [4].
Let G be a finite group. A frame (ϕg )g∈G for H is a group frame or G-frame if there exists a unitary
representation U : G −→ U(H) such that gϕh := U (g)ϕh = ϕgh for all g, h ∈ G. We will consider only
unitary irreducible representations (unirreps).
1. Frame Quantization. Fix a unirrep (U, H) of the finite group G. Consider the following set A ={|U (x)i : x ∈ G} of finite tight group frames on the space B(H) of linear operators on H. Here|U (x)iP= U (x)I ∈ B(H) and I is the identity operator. We have the following frame quantization
F 7→ x,y∈G F (x, y)|U (x)ihU (y)| where F : G × G −→ C.
2. Star-product Formula. Consider the restriction L2 (G × G) onto L2 (G) given by f (x, y0 ) 7→ f (x).
Conversely, if f ∈ L2 (G), a function fe ∈ L2 (G × G) is obtained by fe(x, y0 ) = f (x). Here one
must fix y0Pat the beginning, say y0 = e, the identity of the group. Then, the quantization
bf =
b
b b
f 7→ Ax∈G f (x)|U (x)ihU (y)| provides the following star-product of functions Af ?g = Af ◦ Ag .
The bijection between operators and phase
P space functions gives the formula for the star-product,known from the literature, f ? g(z) = x,y∈G f (x)g(y)T r[U (xyz −1 )] [1].
3. Geometrization. Let U (x) = (uij (x)) be the matrix elements of a unirrep U. From the PeterWeyl Theorem, the functions uij : G −→ C form an orthonormal basis for the restriction of theregular representation to the space equivalent to the unirrep. Then the star-product ? = ?λ has
the following geometrization by contravariant tensors, from which follows also the geometrizationof the Lie and Jordan brackets [3]:




 

 

X ∂
X ∂
∂
∂
∂
∂
∂
∂
 − pkn 

⊗
−
⊗
⊗
−
⊗
? = qkn 
∂q
∂q
∂p
∂p
∂q
∂p
∂q
∂p
nj
jk
nj
jk
nj
jk
nj
jk
j
j







 


X ∂
X
∂
∂
∂
∂
∂
∂
∂
 + iqkn .
+ipkn 
⊗
−
⊗
⊗
−
⊗
∂q
∂q
∂p
∂p
∂q
∂p
∂q
∂p
nj
jk
nj
jk
nj
jk
nj
jk
j
j

References[1] P. Aniello, Star products: a group-theoretical point of view, J. Phys. A: Math. Theor. 42 (2009), 475210–
475239.
[2] P. Casazza, Finite Frames: Theory and Applications, Applied and Numerical Harmonic Analysis, Birkhäuser,
New York, 2013.
[3] C.N. Mañibo, Poisson and Hopf Structures for Finite Groups via Star Product and Convolution, 32nd
Physics Congress of the Samahang Pisika ng Pilipinas, 2014, Quezon City, Philippines, October 17–20, 2014,
pp. SPP2014–3B–01–1.
[4] S. F. D. Waldron, An Introduction to Finite Tight Frames, Applied and Numerical Harmonic Analysis,
Birkhäuser, New York, 2018.
[5] C.K. Zachos, D.B. Fairlie, and T.L. Curtright, Quantum Mechanics in Phase Space, World Scientific Seriesin 20th Century Physics, vol. 34, World Scientific Publishing Co. Pte. Ltd., Singapore, 2005.

125

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Direction-Critical Configurationsin Noncentral-General Position
Silvia Fernández-Merchant1 and Rimma Hämäläinen1
1

Department of Mathematics, California State University, Northridge, 18111 Nordhoff St. Northrige CA
91330-8313, USA
silvia.fernandez@csun.edu (S. Fernández-Merchant),rimma.hamalainen.238@my.csun.edu (R. Hämäläinen)

Abstract
In 1982, Ungar proved that the connecting lines of a set of 2n points in the plane, not all
collinear, determine at least 2n slopes. Sets achieving this minimum are called direction-critical(or slope-critical ). By using allowable sequences, Ungar actually proved this result for generalized
configurations of points (where points can be connected by pseudolines instead of straight lines). In
1986, Jamison proved that any direction-critical configuration of points in general position is an affine
transformation of the vertices of a regular polygon. We study direction-critical centrally symmetric
generalized configurations of points that are in noncentral-general position (where only the connecting(pseudo)lines not through the center of symmetry can pass through more than two points). We provethat, up to equivalence, the numbers of points in the (pseudo)lines through the center of symmetrycompletely determine a centrally symmetric direction-critical generalized configuration of points innoncentral-general position, and a configuration exists for any such set of numbers. We also showthat infinitely many of this direction-critical generalized configurations of points are not stretchable.
Keywords: slope-critical, generalized configuration of points, pseudoline arrangement, allowable sequence, centrally symmetric.
2010 MSC: Primary 52C30; Secondary 05C10, 05C35, 52C10.

1Introduction

In 1970, Scott [6] proposed the problem of finding the least number of directions (slopes) determined bya set of n points in the plane, not all collinear. He conjectured that the minimum number of different
slopes determined by the connecting lines of a set of n points in the plane is 2bn/2c. This conjecture wassettled by Ungar [7] in 1982 using allowable sequences. An allowable sequence of n points is a doublyinfinite periodic sequence of permutations Π = {πi }∞i=−∞ of the set of points {1, 2, 3, . . . , n}, satisfying
the following properties: (1) By relabeling the points, if necessary, it can be assumed that π0 = 123 . . . n.
(2) There is h ∈ Z+ such that πi+h is the reversal of πi for every i ∈ Z. Then Π has period 2h and{π0 , π1 , . . . , πh } is a halfperiod of Π (of length h). (3) πi+1 is obtained from πi by the reversal of one ormore disjoint substrings, each involving consecutive elements of πi . These reversals are called switches.
(4) Any pair of points switches exactly once within a halfperiod.
Allowable sequences were used by Goodman and Pollack to approach combinatorial problems of setsof points in the 1980s [1, 2]. Any set P of n points in the plane determines an allowable sequence Π(P ),called its circular sequence. The permutations of Π(P ) are determined by the projections of P onto a linethat rotates around a circle enclosing P . So every set of points corresponds to an allowable sequence but
not every allowable sequence is the circular sequence of a set of points. In fact, Goodman and Pollack [3]showed that up to combinatorial equivalence, there is a one-to-one correspondence between the set ofallowable sequences and the set of generalized configurations of points, that is, a set of points together
with a pseudoline arrangement so that every pair of points is contained in a unique pseudoline. So
Ungar’s proof actually extends the problem of finding the minimum number of directions from point-sets
to generalized configurations of points. In this new setting, all switches occurring between consecutive
permutations of an allowable sequence correspond to pseudolines determining the same direction. So the126

S. Fernández-Merchant et al. Direction-Critical Configurations in NC-General Position

TJCDCGGG2021

number of directions determined by an allowable sequence Π is the length h(Π) of its halfperiod. Ungar
proved that h(Π) ≥ 2bn/2c. Indeed, the number of directions determined by a set of points P is thesame as the length of a halfperiod of its circular sequence Π(P ). The configurations/allowable sequences
achieving this minimum are called direction-critical or slope-critical.
We consider centrally symmetric point-sets and generalized configurations of points. They correspondto allowable sequences whose permutations are centrally symmetric. We only study allowable sequenceswith an even number of points as every odd centrally symmetric allowable sequence is obtained from aneven one by adding its center of symmetry. The switches reversing a centered substring of a permutation
are called crossing switches. They correspond to connecting lines passing through the center of symmetryof a point-set, which are called central connecting lines. The number of points on each side of the centerof symmetry along a central connecting line, or equivalently, half of the length of the corresponding
crossing switch, is called the distance of the crossing switch. It follows from Ungar’s proof that thesum of the distances d1 , d2 , . . . , dt of all crossing switches in a half period of a direction-critical centrally
symmetric allowable sequence on 2n points is exactly n. The ordered partition n = d1 + d2 + · · · + dt
is called the crossing distance partition of the allowable sequence, where the terms go with the order inwhich the corresponding switches occur within a half period. Jamison [5] proved that all slope-criticalconfigurations of 2n points in general position are affine transformations of the regular 2n-gon. We
relax the general position assumption by allowing connecting lines through the center of symmetry topass through more than two points, but all other connecting lines pass through exactly two points. In
the setting of allowable sequences, this means that all switches, except for the crossing switches, aretranspositions of two points. We say that these sets are in noncentral-general position.
The following problem is considered: For which positive integer vectors (d1 , d2 , . . . , dt ) there are
direction-critical centrally symmetric point-sets or generalized configurations of points in noncentralgeneral position with crossing distance partition n = d1 + d2 + · · · + dt ?

2Main Results

Theorem 2.1. Let d = (d1 , d2 , . . . , dt ) be a vector with positive integer entries. Then, up to combinatorial
equivalence, there is a unique direction-critical centrally symmetric allowable sequence Π(d) in noncentralgeneral position with crossing distance partition n = d1 + d2 + · · · + dt .
Some of these allowable sequences are known to be the circular sequence of a point-set, i.e., geometrically realizable or stretchable. Moreover, due to the uniqueness in Theorem 2.1, the corresponding
point-sets are unique up to affine transformations, as stated in the following corollaries.
Corollary 2.2. Let d = (1, 1, . . . , 1) ∈ Zn . The allowable sequence Π(d) with crossing distance partitionn = 1 + · · · + 1 is uniquely geometrically realized, up to affine equivalence, by the regular 2n-gon.
Corollary 2.3. Let d = (d1 , d2 ). The allowable sequence Π(d) with crossing distance partition n = d1 +d2
is uniquely geometrically realized, up to affine equivalence, by the exponential cross EXλ (d1 − 1, d2 − 1)(without its center) as described by Jamison in [4], or by the bipencil when d1 = 1 or d2 = 1.
However, the allowable sequence Π(d) in Theorem 2.1 is not geometrically realizable for infinitely
many vectors d.
Theorem 2.4. Let d = (d1 , d2 , . . . , dt ) be a vector with positive integer entries. Then the allowablesequence Π(d) is not geometrically realizable whenever (1) at least 3 entries of d are greater than or equal
to 3, or (2) at least 4 entries of d are greater than 1.

References[1] J. E. Goodman and R. Pollack, On the combinatorial classification of nondegenerate configurations in theplane, J. Combin. Theory Ser. A 29 (1980), 220–235.
[2] E. Goodman and R. Pollack, A combinatorial perspective on some problems in geometry, Congressus Numerantium, 32 (1981), 383–394.
[3] E. Goodman and R. Pollack, Semispaces of configurations, cell complexes of arrangements, J. Combin.
Theory Ser. A 37-3 (1984), 257–293.
[4] R.E. Jamison, Planar configurations which determine few slopes, Geometriae Dedicata, 16 (1984) 17–34.
[5] R.E. Jamison, Few Slopes Without Collinearity, Discrete Mathematics 60 (1986), 24-31.
[6] P. R. Scott, On the sets of directions determined by n points, Amer. Math. Monthly 77 (1970), 502–505.
[7] P. Ungar, 2N Noncollinear Points Determine at Least 2N Directions, J. Combin. Theory 33 (1982), 343–347.

127

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Parallel Curves Detection Method Based on Hough Transform
Nattapol Chanpaisit1 , and Pat Vatiwutipong1
1

Kamnoetvidya Science Academy, Rayong 21210, Thailand
pat.v@kvis.ac.th (P. Vatiwutipong), nattapol c@kvis.ac.th (N. Chanpaisit)
Abstract
For recognizing straight lines and circles, the Hough transform is a widely used approach. Recently, an extension was published that allows us to detect a larger class of algebraic curves. The
image with parallel curves is the subject of this study. An incorrect curve may be detected using
the standard approach. We modify the accumulator function and introduce the concept of parallel
curves into the detection method to tackle this problem. Synthetic and real-world images are used
to demonstrate the effectiveness of this new methodology.
Keywords: Parallel curves, Algebraic plane curves, Pattern recognition, Hough transform
2010 MSC: Primary 14Q05; Secondary 68U10, 14H50.

1

Introduction

The Hough transform, defined in [1], considers all conceivable curves and counts how many points on
the image space each curve contains. The counted value for each curve is referred to as an accumulator
function. As a detected curve, the highest value of the accumulator function will be suggested. We focus
on images with parallel curves. The issue is if any points from different curves are accidentally joined,
our detection may fail. Because the accumulator function will reach a peak at that accidental curve if
this case occurs. Take the following figure as an example if we aim to extract the direction of parallelveins in the image of a banana leaf. The Hough transform misled us to the midrib because there are
more points on that line than on each parallel vein. To avoid this kind of incorrect detection, we will
add a parallel assumption to our modified approach.

2Preliminaries

Definitions and notation of family of curve and the Hough transform follow [1–3].
Definition 2.1. Consider a set L ⊂ {1, . . . , t}. We say that a curve Cλ is L−parallel to a curve Cλ0 if
λi = λ0i for all i ∈/ L.
We denote a family of all curve in FSthat are L−parallel to a curve Cλ(0) by FL,λ(0) . A family FL,λ(0)of curves in Rn is said to be a cover if Cλ ∈F (0) Cλ = Rn , and said to be a partition if it is a cover andL,λfor all distinct Cλ , Cλ0 ∈ FL,λ(0) , we have Cλ ∩ Cλ0 = ∅ or Cλ = Cλ0 . Take the family F1 of straight lines
defined by f(r,θ) (x, y) = r − x cos θ − y sin θ as an example. Let (r0 , θ0 ) ∈ R+
0 × [0, 2π). For L = {1}, asubfamily FL,(r0 ,θ0 ) is the set of all straight lines such that the angle between it and the x-axis is equal toθ0 . This subfamily is a partition. For L = {2}, a subfamily FL,(r0 ,θ0 ) is the set of all straight lines suchthat the distance between it and the origin is equal to r0 . This subfamily is not a cover. Another example
is the family F2 of circles defined by f(h,k,r) (x, y) = r − (x − h)2 − (y − k)2 . Let (h0 , k0 , r0 ) ∈ R2 × R+
0.
128

P. Vatiwutipong et al.

Parallel Curves Detection Method

TJCDCGGG2021

For L = {3}, a subfamily CL,(h0 ,k0 ,r0 ) is the set of all circles with fixed center at (h0 , k0 ). This subfamily
is a partition. For L = {1, 2}, a subfamily FL,(h0 ,k0 ,r0 ) is the set of all circles with fixed radius r0 . Thissubfamily is a cover but not a partition.

3

Main Results

After observing the value of the accumulator function, we found that if there are parallel curves in theimage, then the accumulator function in that direction will contain peaks at each position of the curves.
On the other hand, there would have only one very high rise on the accidentally lined up direction. The
following theorem is the conceptual idea for our proposed parallel curves recognition method.
Theorem 3.1. Let F be a family of irreducible curves parametrized by t dimensional tuple. For L ⊂{1, ..., t}, then the following hold.
1. Fix λ(0) . For each λ such that Cλ ∈ FL,λ(0) , the Hough transform ΓP (λ) of the pairs (Cλ , P ) when(0)

P varies on Cλ all pass through a point on set EL,λ(0) = {Λ ∈ Rt : Λi = λifor all i ∈/ L}.

2. Fix λ(0) . If the Hough transform ΓP (λ) of the pairs (Cλ , P ) when P varies on curves Cλ and theHough transform ΓP 0 (λ) of the pairs (Cλ0 , P 0 ) when P 0 varies on curves Cλ0 in F are all intersected
at two points in the set EL,λ(0) , then Cλ is L−parallel to Cλ0 .
The prior procedure of preventing this false detection is to integrate over all directions in L, that isZZ· · · HF |EL,λ(0) (Λ)dΛ1 . . . dΛ`
(3.1)
R

R

where HF is a accumulator function. But if FL,λ(0) is a partition, we can show the integration 3.1 isa constant. As a result, we shall first apply some function F to the integrand rather than integrating
directly. The function should be increasing. Otherwise, the order of the accumulator function will be
lost. Moreover, we should not use linear or affine linear because the integral will remain constant. We
recommend using a simple function such as square, soft thresholding, or hard thresholding function. Our
proposed method is, instead of seeking for the maximum of the accumulator function, we seek for the
peak ofZZ

· · · F HF |EL,λ(0) (Λ) dΛ1 . . . dΛ`
(3.2)
R

R

where F is a function that we mention. That will provide us the direction of the parallel curves. After
that, we project the accumulator function on R` . Then consider peaks of it to obtain the positions on
those parallel curves.
We apply the proposed method to several synthetic inputs. Consider the image of circles of different
sizes with the same center, which are parallel with respect to L = {3}, together with another circle that
contains more points. By using the usual method, the circle with more points will be detected as in the
middle image. On the contrary, our proposed method with the square function will detect parallel circles
as in the right one.

References[1] M. C. Beltrametti, A. M. Massone, and M. Piana, Hough Transform of Special Classes of Curves, SIAM J.
Imaging Sci., 6(1) (2013), 391–412.
[2] G. Ricca, M. C. Beltrametti, and A. M. Massone, Detecting Curves of Symmetry in Images Via Hough
Transform, Math. Comput. Sci. 10 (2016), 179–205.
[3] M. C. Beltrametti, C. Campi, A. M. Massone, and M. Torrente, Geometry of the Hough Transforms with
Applications to Synthetic Data, Math. Comput. Sci. (2020).

129Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Computing the Probability of Striking a BattleshipJack Spalding-Jamieson11David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada
jacksj@uwaterloo.ca (J. Spalding-Jamieson)

Abstract
A set of n non-overlapping rectangular ‘battleships’ with unit width are placed on an H ×W 2D square grid
uniformly at random. Given the state of an in-progress game, and an initial assumption of uniform probability
over all legal configurations of the ships, our goal is to determine the probability that each square contains a ship.
We describe and implement a practical solution to this problem, and extend our implementation to run on GPU.
Keywords: battleship, graph, clique, probability, GPU.
2010 MSC: Primary 68W05; Secondary 91A60, 05C85, 68W10.

1Introduction

Battleship is a game played between two people. Each player starts by placing ‘ships’ into hidden locations on a 2D
square grid unique to their own ships, and then players take turns attempting to ‘attack’ locations on the others’ grid.
In [4], Fiat and Shamir describe an overarching strategy that bounds the worst-case number of attacks required,in the case of one ship. In this paper, we attempt to evaluate local choices within a game of battleship. Our
goal is to quickly compute the exact probability of striking a ship with an attack, for each possible next choice ofsquare point to attack. A method for computing or estimating these probabilities has previously been explored [3],but it is much slower than the techniques described here.

1.1

Probability Queries

We consider only one side in the game of battleship. A defending player places n unit-width ships S ={s1,...,sn}with lengths L={l1,...,ln} onto an H ×W square grid, such that no two ships occupy the same grid square. Notethat we allow the ships to touch, but we will not make special use of this fact. An attacking player is allowed
to perform queries, called attacks on squares, with at most one query per square. Each query on a square r
has three possible responses.
If the attack missed, there is no ship at the sqaure.
If the attack hit, there is a ship at the square, and some part of the ship remains undiscovered.
If the attack sunk, there is a ship at the square, and all parts of the ship have now been discovered. The
specific ship itself is identified in the query response, which we say has been sunk at square r.
The attacking player wishes to minimize the number of queries needed to sink all ships.
In this work we attempt to practically solve the following problem: Assume the ships were placed using auniform distribution over all possible legal configurations of ship placements. Given a series of queries madeby the attacker so far, and their responses, determine the probability that each square of the grid contains a ship.

2

Cliques and the Placement Graph

For a set of ships L and a grid size H ×W , we construct placement graph G1 with the following vertices andedges: For each ship s, and each possible placement p of s onto the grid (a location and orientation), create avertex vs,p. Create an edge for each pair of vertices vs,p,vs0 ,p0 with s6= s0 where s placed at p would not intersectwith s0 placed at p0. The configurations of all ships correspond to n-cliques in G which cover all squares of thegrid that have received hit responses.
Then, given a list of queries and responses so far, we can construct the valid placement subgraph G2
which is an induced subgraph of G1 with the vertices vs,p such that:• No square r that has received a miss response is inside the placement p of s.
• If a ship s has already been sunk at a square r, then vs,p covers only r and squares which received hit
responses prior to the sunk response of r.
130

J. Spalding-Jamieson

Computing the Probability of Striking a Battleship

TJCDCGGG2021

0

Figure 1: Left: A configuration of ships on the 10×10 grid. Right: The same
configuration after some attacks. The 6 × symbols are misses. The 4 empty
circles are hits. The filled circle with a 0 inside is a sink of ship 0. Note thatthe order of the attacks is not specified here.

Figure 2: For a 3×3 board with
L = {2,3}, the above diagram
represents a small subgraph of
G1.

Using the valid placement subgraph, we can compute our probabilities with a simple algorithm. Let h(vs,p)be the number of hit squaress touches, and let H be the total number of hit squares. We iterate through allPn-cliques C of G2. If v∈C h(v) = H, then C corresponds to a valid configuration of all ships given all theinformation we have. For each vertex vs,p, we let fvs,p denote the number of valid configurations containing
the ship s in placement p. Finally, the total number of valid configurations covering a square r is the sum of fvs,pover all placements p of a ship s that overlap with r. This value is sufficient to compute the exact probability
that r appears in a configuration, by dividing by the total number of valid configurations.

3

Clique Iteration and Performance Engineering

The primary performance bottleneck of this approach is the iteration through all n-cliques. Bron and Kerbosch
describe a method for iterating through all maximal cliques of general graphs [2]. However, for our n-partite
graph, this can be done much more quickly with a simple recursive approach: Iterate through all placements
p1 of s1. Inside the iteration of p1, iterate through all placements p2 of s2 which do not conflict with the current
placement of s1. Inside the iteration of p2, iterate through all placements p3 of s3 which do not conflict withthe current placements of s2 or s1. And so on. This pattern can be implemented as a simple recursive function.
An implementation of our simple recursive approach can be refined for much better real-time performance: Foreach ship si, a set Vi of placements of si which do not intersect with the current placements of ships s1,s2,...,si−1
can be maintained. Furthermore, these sets can be updated quickly too: For each pair of ships si and sj with i<j,and every pair of placement pi of si, we can compute a set Bi,pi ,j of placements of j which do not overlap pi. Notethat all such sets can be computed prior to the n-clique iteration, and that in practice the computation of these sets
is negligible. When choosing a placement pi for a ship si, each set Vj for j >i can now be updated by performing theoperation Vj ←Vj ∩Bi,pi ,j By storing each Vi as a bitset, this operation can be performed extremely fast in practice.
This algorithm can also be adapted for a GPU: Pre-compute the first few levels of recursion on CPU, andcompute the remaining levels in parallel on GPU.

4

Experimental Results

We evaluated our implementation on the version of the game with n=5, L={5,4,3,3,2}, and H ×W =10×10.
The input with the most valid configurations is the empty board, so it provides an upper bound on performance.
We tested on an AMD Ryzen 5 1400 CPU and an NVIDIA RTX 3080 GPU. Our basic single-threaded CPU
implementation was able to solve the empty board in 146.37s. Our parallelized implementation using hipSYCL [1]completed in 5.70s on GPU, and in 36.36s using 8 CPU threads.
Acknowledgments. The authors are grateful to the referees for their careful reading of the manuscript andtheir useful comments.

References[1] Aksel Alpay and Vincent Heuveline, Sycl beyond opencl: The architecture, current state and future direction of hipsycl,
Proceedings of the International Workshop on OpenCL (New York, NY, USA), IWOCL ’20, Association for Computing
Machinery, 2020.
[2] Coen Bron and Joep Kerbosch, Algorithm 457: Finding all cliques of an undirected graph, Commun. ACM 16 (1973),
no. 9, 575–577.
[3] C. Liam Brown, Battleship probability calculator: Methodology, https://web.archive.org/web/20210709154432/
https://cliambrown.com/battleship/methodology.php, Accessed: 2021-07-09.
[4] Amos Fiat and Adi Shamir, How to find a battleship, Networks 19 (1989), no. 3, 361–371.

131

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

The Legend of Zelda: The Complexity of Mechanics
Jeffrey Bosboom1 , Josh Brunner1 , Michael Coulombe1 , Erik D. Demaine1 ,Dylan H. Hendrickson1 , Jayson Lynch2 , Lorenzo Najt3
1

Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
2
Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada
3
Department of Mathematics, University of Wisconsin, Madison, WI, USA
jbosboom@mit.edu (J. Bosboom), brunnerj@mit.edu (J. Brunner), mcoulomb@mit.edu (M. Coulomb),edemaine@mit.edu (E. Demaine), dylanhen@mit.edu (D. Hendrickson), jayson.lynch@waterloo.ca (J. Lynch),lnajt@math.wisc.edu (L. Najt)

Abstract
We analyze some of the many game mechanics available to Link in the classic Legend of Zelda
series of video games. In each case, we prove that the generalized game with that mechanic ispolynomial, NP-complete, NP-hard and in PSPACE, or PSPACE-complete. In the process we givean overview of many of the hardness proof techniques developed for video games over the past
decade: the motion-planning-through-gadgets framework, the planar doors framework, the doorsand-buttons framework, the “Nintendo” platform game / SAT framework, and the collectible tokens
and toll roads / Hamiltonicity framework.
Keywords: video games, hardness, NP, PSPACE.
2010 MSC: Primary 68Q17; Secondary 68Q25.

1Introduction

The Legend of Zelda∗ action–adventure video game series consists of 19 main games developed by Nintendo (sometimes jointly with Capcom), starting with the famous 1986 original which sold over 6.5
million copies [6], and most recently with Breath of the Wild which was a launch title for Nintendo
Switch (and is arguably what made the Switch an early success). In each game, the elf protagonist Link
explores a world with enemies and obstacles that can be overcome only by specific collectible items andabilities. Starting with nothing, Link must successively search for items that unlock new areas withfurther items, until he reaches and defeats a final boss enemy Ganon.
Across the 35-year history of the series, many different mechanics have been introduced, leading toa varied landscape of computational complexity problems to study: what is the difficulty of completinga generalized Zelda game with specific sets of items, abilities, and obstacles? Reviewing the two Zelda
wikis† and playing the games ourselves, we have identified over 80 unique items with unique mechanics,
throughout the 19 games in the Zelda franchise.
In tribute to the fun and challenge of the Zelda series, we propose a long-term undertaking where thevideo-game-complexity community throughly catalogs these mechanics and analyzes which combinations
lead to polynomial vs. NP-hard computational problems. Toward this goal, we analyze in this paper thecomplexity of several new combinations of various items. Table 1 summarizes our results, along withpreviously known results about Legend of Zelda [1, 3].
Our new results also serve to highlight different techniques for proving polynomial/NP algorithms,
NP-hardness, and PSPACE-hardness of video games involving the control of a single agent/avatar.
For algorithms, we see the powerful approach of dynamic programming combined with the techniqueof shortcutting. One major category is Hamiltonian Path inspired reductions, often simplified with
Viglietta’s Metatheorem 2 [7] concerning collectible items and toll roads. Next is the Nintendo-style SAT∗ All products, company names, brand names, trademarks, and sprites are properties of their respective owners.
† https://zelda.fandom.com/wiki/Category:Items and https://www.zeldadungeon.net/wiki/Category:Items

132

J. Bosboom et al.

The Legend of Zelda: The Complexity of Mechanics

Game Mechanics
Hookshot, Pots, Pits
Hookshot, Pots, Pits, Keys
Switch Hook, Diamond Blocks, Pits
Crystal Switches, Raised Barriers
Roc’s Feather, Pegasus Seeds
Bombs, Renewing Cracked Walls
Ice Arrows, Water
Fairies, Bottles, Unavoidable
Damage Region
Magic Armor, Unavoidable Damage
RegionBow or Bombs, and Crystal
Switches for Raised Barriers
Colored-tile floor puzzles
Kodongos, low walls, sword
Buzz Blobs, Master Sword
Decayed Guardians, Bombs
Magnetic gloves, metal orbs, ledges,
jump platforms
Cane of Pacci, ground holes, ledges,
tunnels
Magnesis Rune, metal platforms
Statues, Pressure Plates, Doors
Ancient Orbs, Pedestals, Doors
Minecarts
Once-Pushable Blocks
Once-Pushable/Pullable Blocks,
hookshot, chests, pits, tunnels
Keys, Doors, Ledges
Once-Pushable Blocks, Ice
Buttons, Doors, Teleporters, Pits,
Crystal Switches
Spinners

TJCDCGGG2021

Games with Mechanics
ALTTP, LA, PH, ALBW
ALTTP, LA, PH, ALBW
OoA
ALTTP, LA, OoA, OoS, PH, ALBWOoA, OoS, MM
OoT, MM, OoA, OoS, TMC, ST
MM
ALTTP, LA, OoT, MM, OoA, OoS, FS,
TWW, FSA, TMC, TP, PH, ST, SS, ALBW,
BotWResult
∈P
NP-c.
∈P
∈P
NP-c.
NP-c.
NP-c.

Ref
new
new
new
new
new
new
new

NP-c.

new

ALTTP, OoT, TWW, TP

NP-c.

new

ALTTP, LA, OoA, OoS, PH

NP-c.

new

LA, OoA, TMC
ALTTP
ALTTP, LA, OoA, OoS, TMC, ALBW, TFH
BotW

NP-c
NP-hard
NP-hard
NP-hard

new
new
new
new

OoS

PSPACE-c.

new

FPT in duration
PSPACE-c.
PSPACE-c.

new
new
new

PSPACE-c.

new

PSPACE-c.
PSPACE-c.
NP-c.

new
new
[1]

NP-c.

[1]

TMC
BotW
ALTTP, OoT, MM, OoA, OoS, FS, TWW,FSA, TMC, TP, PH, ST, SS, ALBWBotW
OoA, OoS, TMC
Zelda I, LA, OoA, OoS, TMC
ALTTP, LA, OoT, MM, TWW, ALBW

AoL, ALTTP, LA, OoT, MM, OoA, OoS, FS,
NP-c.
TWW, TMC, TP, PH, ST, SS, ALBW
OoT, MM, OoS, TMC, TP, ST
PSPACE-c.

[1][1]

ALTTP, ALBWPSPACE-c.

[1]

OoA, OoSPSPACE-c.

[3]

Table 1: Summary of new and past results about complexity of various Zelda mechanics.
reduction from [1] which later acted as inspiration for the door-opening gadgets in [4]. For PSPACEhardness, we use the door-and-button framework of Forišek [5] and Viglietta [7]. Finally, we use thedoor gadget from [1] which, along with the other previous work, inspired the gadgets framework for thecomplexity of motion-planning problems [2–4] which we also use. As a secondary goal, we hope that
this paper offers a nice sampling of proof techniques showing the hardness infrastructure that have been
built up in recent years.

References[1] G. Aloupis, E. D. Demaine, A. Guo, and G. Viglietta. Classic Nintendo games are (computationally) hard.
Theoretical Computer Science, 586:135–160, 2015. Originally at FUN 2014.
[2] J. Ani, J. Bosboom, E. D. Demaine, Y. Diomidov, D. Hendrickson, and J. Lynch. Walking through doors ishard, even without staircases: Proving PSPACE-hardness via planar assemblies of door gadgets. In Proceedings of the 10th International Conference on Fun with Algorithms (FUN 2020), pages 3:1–3:23, 2020.
[3] E. D. Demaine, I. Grosof, J. Lynch, and M. Rudoy. Computational complexity of motion planning of a robot
through simple gadgets. In Proceedings of the 9th International Conference on Fun with Algorithms (FUN2018), pages 18:1–18:21, 2018.
[4] E. D. Demaine, D. Hendrickson, and J. Lynch. Toward a general theory of motion planning complexity:
Characterizing which gadgets make games hard. In Proceedings of the 11th Conference on Innovations in
Theoreti cal Computer Science (ITCS 2020), pages 62:1–62:42, 2020.
[5] M. Forišek. Computational complexity of two-dimensional platform games. In Proceedings of the 5th International Conference on Fun with Algorithms (FUN 2010), pages 214–227, 2010.
[6] Video Game Sales Wiki. The Legend of Zelda. https://vgsales.fandom.com/wiki/The_Legend_of_Zelda.
[7] G. Viglietta. Gaming is a hard job, but someone has to do it! Theory Comput. Syst., 54(4):595–621, 2014.

133

Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Super Catalan Numbers, Chromogeometry, and Fourier
Summations over Finite Fields
Kevin Limanta and Norman Wildberger
School of Mathematics and Statistics, UNSW Sydney, Australiak.limanta@unsw.edu.au, n.wildberger@unsw.edu.au
Abstract
In this talk, we give an algebraic interpretation of the super Catalan numbers
S(m, n) =(2m)!(2n)!
m!n!(m + n)!

in the context of integrating polynomials over unit circles over finite field. We work in three metrical geometries: the Euclidean geometry and two relativistic geometries and show that the threegeometries are related to each other. This is the phenomenon of chromogeometry.
Keywords: Super Catalan numbers, chromogeometry, Fourier summations, discrete geometry.
2010 MSC: Primary 43A70, 11T06, 51E26.

1Introduction

The Super Catalan numbers are family of integers which, up to date, have no general combinatorialinterpretations although there are some combinatorial interpretations for S(m, n) for a particular valueof m and n (see [3–6]). In this talk, we are able to give an algebraic interpretation of the super Catalan
numbers through its closely related family of rational numbersΩ(m, n) =(2m)!(2n)!
S(m, n)
= m+n
4m+n
4
m!n!(m + n)!

which we will call the circular super Catalan numbers.
We shall see that Ω(m, n) has a central algebraic interpretation in the context of integration orsummation theory of polynomials over unit circles over finite fields of odd characteristic. Their role
extends even to summations over suitably defined unit circles in relativistic geometries.
The context of summing polynomials over circles in the plane over different kinds of metrical geometries is the three-fold symmetry of chromogeometry [2] which is an outgrowth of rational trigonometry [1],and brings together Euclidean planar geometry with quadratic form x2 + y 2 (blue) with two different
forms of relativistic Einstein-Minkowski geometries with quadratic forms x2 − y 2 (red) and xy (green).
The relations and indeed symmetries between these three geometries play a crucial role in also our derivations, as formulas for summations over circles in both Euclidean and relativistic geometries are closely
related, and connections between them provide a powerful tool for explicit evaluations.

2

Preliminaries

Consider the finite field Fq with q = pr elements for some prime p > 2 and r ∈ N and defineΩp (m, n) = Ω(m, n)

mod p.

We note that Ωp (m, n) is well defined since S(m, n) is always an integer and the denominator 4m+n is
not a multiple of p.
Introduce three different metrical geometries: Euclidean (blue) and relativistic (red and green) geometry with respective associated quadratic forms x2 + y 2 , x2 − y 2 , and xy. The blue unit circle Sb,q is134Kevin Limanta et al.

SC Numbers, Chromogeometry, and Fourier Summations over FF

TJCDCGGG2021

defined as the set of (x, y) ∈ F2q such that x2 + y 2 = 1 with the red unit circle Sr,q and green unit circle
Sg,q defined similarly. The subscript b, r, g stand for blue, red, and green respectively. For brevity, we
use the subscript c to indicate an arbitrary color.
There is also a dichotomy of whether −1 is a square or not in Fq and we note that distinction throughthe Jacobi symbol which is a generalization of the Legendre symbol. For any odd prime p, the Legendre
symbol is defined as(1
if −1 is a square in Fq ,−1:=p
−1 if −1 is not a square in Fq
 r
 
:= −1
.
and thus for any power of prime q = pr we define the Jacobi symbol −1qpDefine the Fq -valued summation functionals ψc,q on the space of polynomials in two variables α andβ to be the linear functional whose values on a monomial αk β l for k, l ∈ N are given by
 X

X
X
−1
ψg,q = −xk y l , ψr,q = −xk y l .
xk y l , ψb,q = −q(x,y)∈Sg,q(x,y)∈Sb,q(x,y)∈Sr,q

We will show that these functionals satisfy the following three conditions.
1. (Normalization) For the polynomial 1 which is constant equal to 1, we have ψc,q (1) = 1F .
2. (Locality) For any polynomial f that restricts to zero on Sc,q , we have that ψc,q (f ) = 0.
3. (Invariance) ψc,q is invariant under any linear transformation that preserves the bilinear form of
the associated geometry c.
We call any linear functional that satisfies the three conditions above circular integral functional.

3

Main Results

We state our main results without proof.
Theorem 3.1. The linear functionals ψc,q is the unique circular integral functional in each geometry.
We present a simplified and restricted version of the formulas for ψc,q involving the circular super
Catalan numbers in the red and blue geometry, whose derivation stems from the result in the green
geometry.
Theorem 3.2. Let a finite field Fq be given where q = pr .
1. In the red geometry, for 0 ≤ k + l < q − 1,((−1)n Ωp (m, n)ψr,q αk β l =02. In the blue geometry, for 0 ≤ k + l < q − 1,(
Ωp (m, n)k lψr,q α β =0

if k = 2m and l = 2n,otherwise.

if k = 2m and l = 2n,otherwise.

References[1] N. Wildberger, Divine Proportions: Rational Trigonometry to Universal Geometry, Wild Egg Books, Sydney,
2005.
[2] N. Wildberger, Chromogeometry, Mathematical Intelligencer. 32 (2010), 26–32.
[3] E. Allen and I. Gheorghiciuc, A weighted interpretation for the super Catalan numbers, Journal of Integer
Sequences 17.3 (2014).
[4] X. Chen and J. Wang, The Super Catalan Numbers S(m, m + s) for s ≤ 4 arXiv preprint arXiv:1208.4196(2012).
[5] I. Gessel and G. Xin, A combinatorial interpretation of the numbers 6(2n)!/n!(n + 2)!, Journal of Integer
Sequences 8, 05.2.3 (2005).
[6] I. Gheorghiciuc and G. Orelowitz, Super-Catalan Numbers of the Third and Fourth Kind, arXiv preprint
arXiv:2008.00133v1 (2020).

135Extended Abstract
The 23rd Thailand-Japan Conference on
Discrete and Computational Geometry, Graphs, and Games
Department of Mathematics, Chiang Mai University

Snakes and Ladders with Large Spinners
under an Alternative Winning Rule
Waitin Sinthu-urai1 , and Piyashat Sripratak1
1

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
waitin.sint@gmail.com (W. Sinthu-urai), psripratak@gmail.com (P. Sripratak)

Abstract
Snakes and Ladders is a game that players move from square to square to reach the last one. On
each turn, one player spins a spinner that gives a number m ∈ {1, 2, . . . , r} with equal probability,
and moves forward for m squares. If a player spins a number that exceeds the number of theremaining squares, the player cannot move in that turn. However, under an alternative winning
rule in Thailand, the player keeps going until reaching the last square and moves backward with theremaining number.
We focus on the expected number of turns until a player exactly reaches the last square of ann-square board when the value of r is close to n. If there are no ladders leading to the last square,
the cases r = n and r = n − 1 give the same expected number n, which is different from the casewhen r = n − 2 whose expected number is less than n if there are no snakes sending a player backto square 1. We provide an approach to calculate the expected numbers for the case when there isa single ladder leading to the last square for r = n and r = n − 1.
Keywords: board game, expected number, snakes and ladders.
2010 MSC: Primary 60C05.

1 Introduction
Snakes and Ladders or Chutes and Ladders is a game that players take turns to walk from square tosquare following the number from a spinner that provides a number in {1, 2, . . . , r} with equal probability.
This number r is called the range of the spinner. The winner is the player who exactly stops at the lastsquare. Moreover, there are ladders which lead the player who stops at the lower square up to the upper
one. On the other hand, there are snakes which drop the player who stops at the upper square down
to the lower one. If the player is so close to the last square that the number gotten from the spinner ismore than the number of the remaining squares, that player cannot walk and has to wait until the next
turn. However, in Thailand, the winning rule is different. If this situation occurs, that player has towalk forward until reaching the last square, and walk backward from the last square with the number ofsteps left.
The expected number of turns for a player to win this classical game is a topic investigated by manymathematicians. Most of them follow the regular winning rule and apply Markov chain model (Seeeg. [1, 3, 4]). Among these works, the study by Connors and Glass [2] considered the case when theranges of spinners are large, and dealt with the problem in a more combinatorial way. Whether using
the combinatorial approach as in [2] or Markov chain techniques, the alternative rule in Thailand makes
it more complicated to calculate the expected number. Therefore, we focus on the expected number of
spins to win the game using large spinners under this alternative winning rule.
There are some properties that hold for most of the boards played in these days, and we assume thesefollowing properties throughout this study.
(i) Each snake’s head is in the upper square and its tail is in the lower one.
(ii) There are no snake’s heads in the last square.
(iii) There are no snakes or ladders before square 1.
(iv) In a square, there exists at most one of these: a snake’s head, a snake’s tail or an end of a ladder.
136

W. Sinthu-urai et al.

Snakes and Ladders under an Alternative RuleTJCDCGGG2021

2 Main Results
Let E denote the expected number of spins to win the game on an n-square board under a given condition.
The first case in our study focuses on the boards with no ladders leading to the last square. To findthe expected number E in this case, we consider the case that there are no snakes or ladders using thefollowing lemma.
Theorem 2.1. [5] Consider a sequence of independent trials that each of them has two outcomes,
success with probability p and failure with probability 1 − p. The expected number of trials until the firstsuccess is p1 .
Since adding any snakes or ladders does not affect the probability of exactly reaching the last squarein each trial, as long as no ladders leading to the last square, the results for this specific case lead tothese theorems.
Theorem 2.2. For an n-square board in the first case, if r = n or r = n − 1, then E = n.
Theorem 2.3. For an n-square board in the first case, if r = n − 2 and there are no snakes sending aplayer back to square 1, then E < n.
As for the second case when there is a single ladder leading to the last square, we let Ei be theexpected number of spins to win the game when we are at square i and C be the coeﬀicient matrix ofthe system CE = 1n−2 where 1n−2 is the (n − 2)-dimension vector whose components are all 1 and[]T
E = E1 · · · Ek−1 Ek+1 · · · En−1 .
For r = n and r = n − 1, we show that for each case, C is invertible and can be used to calculate
E1 , · · · , Ek−1 , Ek+1 , · · · , En−1 . Finally, we can compute the expected number E. We summarize ourworks for the n-square boards in this case as follows.
1∑
Ei where each Ei can be obtained from E = C −1 1n−2 .
n i=1
n−1

Theorem 2.4. If r = n, then E =i̸=kTheorem 2.5. If r = n − 1, then E =n−1
1 ∑Ei where each Ei can be obtained from E = C −1 1n−2 .
n − 1 i=1i̸=kAcknowledgments. The authors are grateful to the Institute for the Promotion of Teaching Scienceand Technology (IPST) for giving us a scholarship under the Development and Promotion of Scienceand Technology Talents (DPST) Project.

References[1] S. C. Althoen, L. King and K. Schilling, How long is a game of snakes and ladders?, The Mathematical
Gazette 77 (1993), 71–76.
[2] D. Connors and D. Glass, Chutes and ladders with large spinners, The College Mathematics Journal 45(2014), 289–295.
[3] L. A. Cheteyan, S. Hengeveld and M. A. Jones, Chutes and ladders for the impatient, The College Mathematics Journal 42 (2011), 2–8.
[4] D. E. Daykin, J. E. Jeacocke and D. G. Neal, Markov chains and snakes and ladders, The Mathematical
Gazette 51 (1967), 313–317.
[5] C. Stein, R. Drysdale and K. Bogart, Discrete Mathematics for Computer Scientists, Addison-Wesley, 2011.

137