bluuebunny/crossref_metadata_2025 huggingface dataset:
New Constructions of Self-Complementary Cayley Graphs
Authors: Wang Lei , Li Cai Heng , Liu Yin , Wu Ci Xuan
Published: January 2020

Self-complementary Cayley graphs are useful in the study of Ramsey numbers, but they are relatively very rare and hard to construct. In this paper, we construct several families of new self-complementary Cayley graphs of order $p^4$ where $p$ is a prime and congruent to $1$ modulo $8$.
On Parking Functions and the Zeta Map in Types B, C and D
Authors: Sulzgruber Robin , Thiel Marko
Published: January 2020

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.
Bijection Between Oriented Maps and Weighted Non-Oriented Maps
Authors: Czyżewska-Jankowska Agnieszka , Śniady Piotr
Published: January 2020

We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non-oriented map is counted with a multiplicity which is based on the concept of the orientability generating series and the measure of orientability of a map. This bijection has the remarkable property of preserving the underlying bicolored graph. Our bijection shows equivalence between two explicit formulas for the top-degree of Jack characters, i.e. (suitably normalized) coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions.
On $t$-Common List-Colorings
Authors: Choi Hojin , Kwon Young Soo
Published: January 2020

In this paper, we introduce a new variation of list-colorings. For a graph $G$  and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) =  \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.
Toggling Independent Sets of a Path Graph
Authors: Joseph Michael , Roby Tom
Published: January 2020

This paper explores the orbit structure and homomesy (constant averages over orbits) properties of certain actions of toggle groups on the collection of independent sets of a path graph. In particular we prove a generalization of a homomesy conjecture of Propp that for the action of a "Coxeter element" of vertex toggles, the difference of indicator functions of symmetrically-located vertices is 0-mesic. Then we use our analysis to show facts about orbit sizes that are easy to conjecture but nontrivial to prove. Besides its intrinsic interest, this particular combinatorial dynamical system is valuable in providing an interesting example of (a) homomesy in a context where large orbit sizes make a cyclic sieving phenomenon unlikely to exist, (b) the use of Coxeter theory to greatly generalize the set of actions for which results hold, and (c) the usefulness of Striker's notion of generalized toggle groups.
Nearly Neighbourly Families of Standard Boxes
Authors: Bojarski Jacek , Kisielewicz Andrzej Piotr , Przesławski Krzysztof
Published: January 2020

It is shown that each nearly neighbourly family of standard boxes in $\mathbb{R}^3$ has at most 12 elements. A combinatorial classification of all such families that have exactly 12 elements is given. All the families satisfying an extra property called incompressibility are described. Compressible families are discussed briefly.
Analysis of the Gift Exchange Problem
Authors: Apagodu Moa , Applegate David , Sloane N. J. A. , Zeilberger Doron
Published: January 2020

In the gift exchange game there are $n$ players and $n$ wrapped gifts. When a player's number is called, that person can either choose one of the remaining wrapped gifts, or can "steal" a gift from someone who has already unwrapped it, subject to the restriction that no gift can be stolen more than a total of $\sigma$ times. The problem is to determine the number of ways that the game can be played out, for given values of $\sigma$ and $n$. Formulas and asymptotic expansions are given for these numbers. This work was inspired in part by a 2005 remark by Robert A. Proctor in the On-Line Encyclopedia of Integer Sequences.This is a sequel to the earlier article [arXiv:0907.0513] by the second and third authors, differing from it in that there are two additional authors and several new theorems, including the resolution of most of the conjectures, and the extensive tables have been omitted.
The Maximum Independent Sets of de Bruijn Graphs of Diameter 3
Authors: Cartwright Dustin A. , Cueto María Angélica , Tobis Enrique A.
Published: January 2020

The nodes of the de Bruijn graph $B(d,3)$ consist of all strings of length $3$, taken from an alphabet of size $d$, with edges between words which are distinct substrings of a word of length $4$. We give an inductive characterization of the maximum independent sets of the de Bruijn graphs $B(d,3)$ and for the de Bruijn graph of diameter three with loops removed, for arbitrary alphabet size. We derive a recurrence relation and an exponential generating function for their number. This recurrence allows us to construct exponentially many comma-free codes of length 3 with maximal cardinality.
Strengthened Brooks' Theorem for Digraphs of Girth at least Three
Authors: Harutyunyan Ararat , Mohar Bojan
Published: January 2020

Brooks' Theorem states that a connected graph $G$ of maximum degree $\Delta$ has chromatic number at most $\Delta$, unless $G$ is an odd cycle or a complete graph. A result of Johansson shows that if $G$ is triangle-free, then the chromatic number drops to $O(\Delta / \log \Delta)$. In this paper, we derive a weak analog for the chromatic number of digraphs. We show that every (loopless) digraph $D$ without directed cycles of length two has chromatic number $\chi(D) \leq (1-e^{-13}) \tilde{\Delta}$, where $\tilde{\Delta}$ is the maximum geometric mean of the out-degree and in-degree of a vertex in $D$, when $\tilde{\Delta}$ is sufficiently large. As a corollary it is proved that there exists an absolute constant $\alpha < 1$ such that $\chi(D) \leq \alpha (\tilde{\Delta} + 1)$ for every $\tilde{\Delta} > 2$.
Toric Mutations in the dP$_2$ Quiver and Subgraphs of the dP$_2$ Brane Tiling
Authors: Gao Yibo , Li Zhaoqi , Vuong Thuy-Duong , Yang Lisa
Published: January 2020

Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we study the del Pezzo 2 (dP$_2$) quiver and its associated brane tiling which arise in theoretical physics. Specifically, we prove explicit formulas for all cluster variables generated by toric mutation sequences of the dP$_2$ quiver. Moreover, we associate a subgraph of the dP$_2$ brane tiling to each toric cluster variable such that the sum of weighted perfect matchings of the subgraph equals the Laurent polynomial of the cluster variable.
Finite Groups of Derangements on the $n$-Cube II
Authors: Cusick Larry , Vega Oscar
Published: January 2020

Given $k\in \mathbb{N}$ and a finite group $G$, it is shown that $G$ is isomorphic to a subgroup of the group of symmetries of some $n$-cube in such a way that $G$ acts freely on the set of $k$-faces, if and only if, $\gcd(k, |G|)=2^s$ for some non-negative integer $s$. The proof of this result is existential but does give some ideas on what $n$ could be.
Asymptotic Behavior of Odd-Even Partitions
Authors: Jang Min-Joo
Published: January 2020

Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's "Lost" Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions. Nonetheless, Andrews found out that this function possesses combinatorial information, odd-even partition. In this paper, we provide the asymptotic formula for this combinatorial object. We also study its companion odd-even overpartitions.
Fixed Points of the Evacuation of Maximal Chains on Fuss Shapes
Authors: Eu Sen-Peng , Fu Tung-Shan , Hsu Hsiang-Chun , Huang Yu-Pei
Published: January 2020

For a partition $\lambda$ of an integer, we associate $\lambda$ with a slender poset $P$ the Hasse diagram of which resembles the Ferrers diagram of $\lambda$. Let $X$ be the set of maximal chains of $P$. We consider Stanley's involution $\epsilon:X\rightarrow X$, which is extended from Schützenberger's evacuation on linear extensions of a finite poset. We present an explicit characterization of the fixed points of the map $\epsilon:X\rightarrow X$ when $\lambda$ is a stretched staircase or a rectangular shape. Unexpectedly, the fixed points have a nice structure, i.e., a fixed point can be decomposed in half into two chains such that the first half and the second half are the evacuation of each other. As a consequence, we prove anew Stembridge's $q=-1$ phenomenon for the maximal chains of $P$ under the involution $\epsilon$ for the restricted shapes.
Rational Dyck Paths in the Non Relatively Prime Case
Authors: Gorsky Eugene , Mazin Mikhail , Vazirani Monica
Published: January 2020

We study the relationship between rational slope Dyck paths and invariant subsets of $\mathbb{Z},$ extending the work of the first two authors in the relatively prime case. We also find a bijection between $(dn,dm)$–Dyck paths and $d$-tuples of $(n,m)$-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and the geometry of affine Springer fibers and knot invariants in the non relatively prime case.
Disjoint Cycles of Different Lengths in Graphs and Digraphs
Authors: Bensmail Julien , Harutyunyan Ararat , Le Ngoc Khang , Li Binlong , Lichiardopol Nicolas
Published: January 2020

In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider other cases such as when the graph is triangle-free, or the $k$ cycles are required to have different lengths modulo some value $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for some regular digraphs and digraphs of small order.
Bispindles in Strongly Connected Digraphs with Large Chromatic Number
Authors: Cohen Nathann , Havet Frédéric , Lochet William , Lopes Raul
Published: January 2020

A $(k_1+k_2)$-bispindle is the union of $k_1$ $(x,y)$-dipaths and $k_2$ $(y,x)$-dipaths, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. showed that for every $(1,1)$- bispindle $B$, there exists an integer $k$ such that every strongly connected digraph with chromatic number greater than $k$ contains a subdivision of $B$. We investigate generalizations of this result by first showing constructions of strongly connected digraphs with large chromatic number without any $(3,0)$-bispindle or $(2,2)$-bispindle. We then consider $(2,1)$-bispindles. Let $B(k_1,k_2;k_3)$ denote the $(2,1)$-bispindle formed by three internally disjoint dipaths between two vertices $x,y$, two $(x,y)$-dipaths, one of length $k_1$ and the other of length $k_2$, and one $(y,x)$-dipath of length $k_3$. We conjecture that for any positive integers $k_1, k_2,k_3$, there is an integer $g(k_1,k_2,k_3)$ such that every strongly connected digraph with chromatic number greater than $g(k_1,k_2,k_3)$ contains a subdivision of $B(k_1,k_2;k_3)$. As evidence, we prove this conjecture for $k_2=1$ (and $k_1, k_3$ arbitrary).
Majority Choosability of Digraphs
Authors: Anholcer Marcin , Bosek Bartłomiej , Grytczuk Jarosław
Published: January 2020

A majority coloring of a digraph is a coloring of its vertices such that for each vertex $v$, at most half of the out-neighbors of $v$ have the same color as $v$. A digraph $D$ is majority $k$-choosable if for any assignment of lists of colors of size $k$ to the vertices there is a majority coloring of $D$ from these lists. We prove that every digraph is majority $4$-choosable. This gives a positive answer to a question posed recently by Kreutzer, Oum, Seymour, van der Zypen, and Wood (2017). We obtain this result as a consequence of a more general theorem, in which majority condition is profitably extended. For instance, the theorem implies also that every digraph has a coloring from arbitrary lists of size three, in which at most $2/3$ of the out-neighbors of any vertex share its color. This solves another problem posed by the same authors, and supports an intriguing conjecture stating that every digraph is majority $3$-colorable.
Fair Representation in the Intersection of Two Matroids
Authors: Aharoni Ron , Berger Eli , Kotlar Dani , Ziv Ran
Published: January 2020

Mysteriously, hypergraphs that are the intersection of two matroids behave in some respects almost as well as one matroid. In the present paper we study one such phenomenon - the surprising ability of the intersection of two matroids to fairly represent the parts of a given partition of the ground set. For a simplicial complex $\mathcal{C}$ denote by $\beta(\mathcal{C})$ the minimal number of edges from $\mathcal{C}$ needed to cover the ground set. If $\mathcal{C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in \mathcal{C}$ meeting each $A_i$ in at least $\lfloor \frac{|A_i|}{\beta(\mathcal{C})}\rfloor$ elements. We conjecture that a slightly weaker result is true for the intersection of two matroids: if $\mathcal{D}=\mathcal{P}\cap \mathcal{Q}$, where $\mathcal{P},\mathcal{Q}$ are matroids on the same ground set $V$ and $\beta(\mathcal{P}), \beta(\mathcal{Q}) \le k$, then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in \mathcal{D}$ meeting each $A_i$ in at least $\frac{1}{k}|A_i|-1$ elements. We prove that if $m=2$ (meaning that the partition is into two sets) there is a set belonging to $\mathcal{D}$ meeting each $A_i$ in at least $(\frac{1}{k}-\frac{1}{|V|})|A_i|-1$ elements.
Composability of Permutation Classes
Authors: Karpilovskij Mark
Published: January 2020

We define the operation of composing two hereditary classes of permutations using the standard composition of permutations as functions and we explore properties and structure of permutation classes considering this operation. We mostly concern ourselves with the problem of whether permutation classes can be composed from their proper subclasses. We provide examples of classes which can be composed from two proper subclasses, classes which can be composed from three but not from two proper subclasses and classes which cannot be composed from any finite number of proper subclasses.
Simplicial Complexes are Game Complexes
Authors: Faridi Sara , Huntemann Svenja , Nowakowski Richard J.
Published: January 2020

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known parameters of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.
Vertex-Transitive Direct Products of Graphs
Authors: Hammack Richard H. , Imrich Wilfried
Published: January 2020

It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.
Convolution Estimates and Number of Disjoint Partitions
Authors: Ivanisvili Paata
Published: January 2020

Let $X$ be a finite collection of sets. We count the number of ways a disjoint union of $n-1$ subsets in $X$ is a set in $X$, and estimate the number from above by $|X|^{c(n)}$ where $$c(n)=\left(1-\frac{(n-1)\ln (n-1)}{n\ln n} \right)^{-1}.$$ This extends the recent result of Kane-Tao, corresponding to the case $n=3$ where $c(3)\approx 1.725$, to an arbitrary finite number of disjoint $n-1$ partitions.
Expanders with Superquadratic Growth
Authors: Balog Antal , Roche-Newton Oliver , Zhelezov Dmitry
Published: January 2020

We prove several expanders with exponent strictly greater than $2$. For any finite set $A \subset \mathbb R$, we prove the following six-variable expander results:$|(A-A)(A-A)(A-A)| \gg \frac{|A|^{2+\frac{1}{8}}}{\log^{\frac{17}{16}}|A|},$$\left|\frac{A+A}{A+A}+\frac{A}{A}\right| \gg \frac{|A|^{2+\frac{2}{17}}}{\log^{\frac{16}{17}}|A|},$ $\left|\frac{AA+AA}{A+A}\right| \gg \frac{|A|^{2+\frac{1}{8}}}{\log |A|},$ $\left|\frac{AA+A}{AA+A}\right| \gg \frac{|A|^{2+\frac{1}{8}}}{\log |A|}.$
Burling Graphs, Chromatic Number, and Orthogonal Tree-Decompositions
Authors: Felsner Stefan , Joret Gwenaël , Micek Piotr , Trotter William T. , Wiechert Veit
Published: January 2020

A classic result of Asplund and Grünbaum states that intersection graphs of axis-aligned rectangles in the plane are $\chi$-bounded. This theorem can be equivalently stated in terms of path-decompositions as follows: There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph that has two path-decompositions such that each bag of the first decomposition intersects each bag of the second in at most $k$ vertices has chromatic number at most $f(k)$. Recently, Dujmović, Joret, Morin, Norin, and Wood asked whether this remains true more generally for two tree-decompositions. In this note we provide a negative answer: There are graphs with arbitrarily large chromatic number for which one can find two tree-decompositions such that each bag of the first decomposition intersects each bag of the second in at most two vertices. Furthermore, this remains true even if one of the two decompositions is restricted to be a path-decomposition. This is shown using a construction of triangle-free graphs with unbounded chromatic number due to Burling, which we believe should be more widely known.
Orbits of Antichains in Certain Root Posets
Authors: Dong Chao-Ping , Wang Suijie
Published: January 2020

Building everything from scratch, we give another proof of Propp and Roby's theorem saying that the average antichain size in any reverse operator orbit of the poset $[m]\times [n]$ is $\frac{mn}{m+n}$. It is conceivable that our method should work for other situations. As a demonstration, we show that the average size of antichains in any reverse operator orbit of $[m]\times K_{n-1}$  equals $\frac{2mn}{m+2n-1}$. Here $K_{n-1}$ is the minuscule poset $[n-1]\oplus ([1] \sqcup [1]) \oplus [n-1]$. Note that $[m]\times [n]$ and $[m]\times K_{n-1}$ can be  interpreted as sub-families of certain root posets. We guess these root posets should provide a unified setting to exhibit the homomesy phenomenon defined by Propp and Roby.
A Combinatorial Proof of a Formula of Biane and Chapuy
Authors: Chewi Sinho , Anantharam Venkat
Published: January 2020

Let $G$ be a simple strongly connected weighted directed graph. Let $\mathcal{G}$ denote the spanning tree graph of $G$. That is, the vertices of $\mathcal{G}$ consist of the directed rooted spanning trees on $G$, and the edges of $\mathcal{G}$ consist of pairs of trees $(t_i, t_j)$ such that $t_j$ can be obtained from $t_i$ by adding the edge from the root of $t_i$ to the root of $t_j$ and deleting the outgoing edge from $t_j$. A formula for the ratio of the sum of the weights of the directed rooted spanning trees on $\mathcal{G}$ to the sum of the weights of the directed rooted spanning trees on $G$ was recently given by Biane and Chapuy. Our main contribution is an alternative proof of this formula, which is both simple and combinatorial.
Defective 3-Paintability of Planar Graphs
Authors: Gutowski Grzegorz , Han Ming , Krawczyk Tomasz , Zhu Xuding
Published: January 2020

A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex is uncolored and has $k$ tokens. In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.
On the Existence of Frobenius Digraphical Representations
Authors: Spiga Pablo
Published: January 2020

A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A digraphical, respectively graphical, Frobenius representation, DFR and GFR for short, of a Frobenius group $F$ is a digraph, respectively graph, whose automorphism group as a group of permutations of the vertex set is $F$. The problem of classifying which Frobenius groups admit a DFR and GFR has been proposed by Mark Watkins and Thomas Tucker and is a natural extension of the problem of classifying which groups that have a digraphical, respectively graphical, regular representation.In this paper, we give a partial answer to a question of Mark Watkins and Thomas Tucker concerning Frobenius representations: "All but finitely many Frobenius groups with a given Frobenius complement have a DFR".
On Vertex-Disjoint Paths in Regular Graphs
Authors: Han Jie
Published: January 2020

Let $c\in (0, 1]$ be a real number and let $n$ be a sufficiently large integer. We prove that every $n$-vertex $c n$-regular graph $G$ contains a collection of $\lfloor 1/c \rfloor$ paths whose union covers all but at most $o(n)$ vertices of $G$. The constant $\lfloor 1/c \rfloor$ is best possible when $1/c\notin \mathbb{N}$ and off by $1$ otherwise. Moreover, if in addition $G$ is bipartite, then the number of paths can be reduced to $\lfloor 1/(2c) \rfloor$, which is best possible.
Symmetric Chain Decompositions of Products of Posets with Long Chains
Authors: David Stefan , Spink Hunter , Tiba Marius
Published: January 2020

We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is taut, i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \cdots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions — the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symmetric chain decompositions (2017), making progress on a conjecture of Shearer and Kleitman (1979). In general, we show that for a finite graded poset $P$, there exists a canonical bijection between symmetric chain decompositions of $P \times m$ and $P \times n$ for $m, n\ge rk(P) + 1$, that preserves the existence of taut chains. If $P$ has a unique maximal and minimal element, then we also produce a canonical $(rk(P) +1)$ to $1$ surjection from symmetric chain decompositions of $P \times (rk(P) + 1)$ to symmetric chain decompositions of $P \times rk(P)$ which sends decompositions with taut chains to decompositions with taut chains.
Hypertopes with Tetrahedral Diagram
Authors: Catalano Domenico , Fernandes M. Elisa , Hubard Isabel , Leemans Dimitri
Published: January 2020

In this paper we construct an infinite family of hypertopes of rank four having the complete graph $K_4$ as diagram. Their group of rotational symmetries is isomorphic to $PSL(2,q)$. It turns out some elements of this family are regular hypertopes and some are chiral. Moreover, we show that the chiral ones have both improper and proper correlations simultaneously.
Shellability of Posets of Labeled Partitions and Arrangements Defined by Root Systems
Authors: Delucchi Emanuele , Girard Noriane , Paolini Giovanni
Published: January 2020

We prove that the posets of connected components of intersections of toric and elliptic arrangements defined by root systems are EL-shellable and we compute their homotopy type. Our method rests on Bibby's description of such posets by means of "labeled partitions": after giving an EL-labeling and counting homology chains for general posets of labeled partitions, we obtain the stated results by considering the appropriate subposets.
Strengthening $(a,b)$-Choosability Results to $(a,b)$-Paintability
Authors: Mahoney Thomas
Published: January 2020

Let $a,b\in\mathbb{N}$. A graph $G$ is $(a,b)$-choosable if for any list assignment $L$ such that $|L(v)|\ge a$, there exists a coloring in which each vertex $v$ receives a set $C(v)$ of $b$ colors such that $C(v)\subseteq L(v)$ and $C(u)\cap C(w)=\emptyset$ for any $uw\in E(G)$. In the online version of this problem, on each round, a set of vertices allowed to receive a particular color is marked, and the coloring algorithm chooses an independent subset of these vertices to receive that color. We say $G$ is $(a,b)$-paintable if when each vertex $v$ is allowed to be marked $a$ times, there is an algorithm to produce a coloring in which each vertex $v$ receives $b$ colors such that adjacent vertices receive disjoint sets of colors.We show that every odd cycle $C_{2k+1}$ is $(a,b)$-paintable exactly when it is $(a,b)$-chosable, which is when $a\ge2b+\lceil b/k\rceil$. In 2009, Zhu conjectured that if $G$ is $(a,1)$-paintable, then $G$ is $(am,m)$-paintable for any $m\in\mathbb{N}$. The following results make partial progress towards this conjecture. Strengthening results of Tuza and Voigt, and of Schauz, we prove for any $m \in \mathbb{N}$ that $G$ is $(5m,m)$-paintable when $G$ is planar. Strengthening work of Tuza and Voigt, and of Hladky, Kral, and Schauz, we prove that for any connected graph $G$ other than an odd cycle or complete graph and any $m\in\mathbb{N}$, $G$ is $(\Delta(G)m,m)$-paintable.
$(q,t)$-Characters of Kirillov-Reshetikhin Modules of Type $A_r$ as Quantum Cluster Variables
Authors: Turmunkh Bolor
Published: January 2020

Nakajima (2003) introduced a $t$-deformation of $q$-characters, $(q,t)$-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima $(q,t)$-characters of Kirillov-Reshetikhin modules satisfy a $t$-deformed $T$-system. The $T$-system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, depending on the choice of direction of evolution. In this paper, we show that the Nakajima $t$-deformed $T$-system of type $A_r$ forms a quantum mutation relation in a quantization of exactly one of the cluster algebra structures attached to the $T$-system.
Disproof of the List Hadwiger Conjecture
Authors: Barát János , Joret Gwenaël , Wood David R.
Published: January 2020

The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$.
The Extremal Function and Colin de Verdière Graph Parameter
Authors: McCarty Rose
Published: January 2020

The Colin de Verdière parameter $\mu(G)$ is a minor-monotone graph parameter with connections to differential geometry. We study the conjecture that for every integer $t$, if $G$ is a graph with at least $t$ vertices and $\mu(G) \leq t$, then $|E(G)| \leq t|V(G)|-\binom{t+1}{2}$. We observe a relation to the graph complement conjecture for the Colin de Verdière parameter and prove the conjectured edge upper bound for graphs $G$ such that either $\mu(G) \leq 7$, or $\mu(G) \geq |V(G)|-6$, or the complement of $G$ is chordal, or $G$ is chordal.
Generalized Non-Crossing Partitions and Buildings
Authors: Heller Julia , Schwer Petra
Published: January 2020

For any finite Coxeter group $W$ of rank $n$ we show that the order complex of the lattice of non-crossing partitions $\mathrm{NC}(W)$ embeds as a chamber subcomplex into a spherical building of type $A_{n-1}$. We use this to give a new proof of the fact that the non-crossing partition lattice in type $A_n$ is supersolvable for all $n$. Moreover, we show that in case $B_n$, this is only the case if $n<4$. We also obtain a lower bound on the radius of the Hurwitz graph $H(W)$ in all types and re-prove that in type $A_n$ the radius is $\binom{n}{2}$. A Corrigendum for this paper was added on May 17, 2018.
Pentagon Contact Representations
Authors: Felsner Stefan , Schrezenmaier Hendrik , Steiner Raphael
Published: January 2020

Representations of planar triangulations as contact graphs of a set of internally disjoint homothetic triangles or of a set of internally disjoint homothetic squares have received quite some attention in recent years. In this paper we investigate representations of planar triangulations as contact graphs of a set of internally disjoint homothetic pentagons. Surprisingly such a representation exists for every triangulation whose outer face is a $5$-gon. We relate these representations to five color forests. These combinatorial structures resemble Schnyder woods and transversal structures, respectively. In particular there is a bijection to certain $\alpha$-orientations and consequently a lattice structure on the set of five color forests of a given graph. This lattice structure plays a role in an algorithm that is supposed to compute a contact representation with pentagons for a given graph. Based on a five color forest the algorithm builds a system of linear equations and solves it, if the solution is non-negative, it encodes distances between corners of a pentagon representation. In this case the representation is constructed and the algorithm terminates. Otherwise negative variables guide a change of the five color forest and the procedure is restarted with the new five color forest. Similar algorithms have been proposed for contact representations with homothetic triangles and with squares.
Convex Geometries are Extremal for the Generalized Sauer-Shelah Bound
Authors: Chornomaz Bogdan
Published: January 2020

The Sauer-Shelah lemma provides an exact upper bound on the size of set families with bounded Vapnik-Chervonekis dimension. When applied to lattices represented as closure systems, this lemma outlines a class of extremal lattices obtaining this bound. Here we show that the Sauer-Shelah bound can be easily generalized to arbitrary antichains, and extremal objects for this generalized bound are exactly convex geometries. We also show that the problem of classification of antichains admitting such extremal objects is NP-complete.
Baxter Posets
Authors: Meehan Emily
Published: January 2020

We define a family of combinatorial objects, which we call Baxter posets.  We prove that Baxter posets are counted by the Baxter numbers by showing that they are the adjacency posets of diagonal rectangulations.  Given a diagonal rectangulation, we describe the cover relations in the associated Baxter poset.  Given a Baxter poset, we describe a method for obtaining the associated Baxter permutation and the associated twisted Baxter permutation.
Erdős-Ginzburg-Ziv Constants by Avoiding Three-Term Arithmetic Progressions
Authors: Fox Jacob , Sauermann Lisa
Published: January 2020

For a finite abelian group $G$, The Erdős-Ginzburg-Ziv constant $\mathfrak{s}(G)$ is the smallest $s$ such that every sequence of $s$ (not necessarily distinct) elements of $G$ has a zero-sum subsequence of length $\operatorname{exp}(G)$. For a prime $p$, let $r(\mathbb{F}_p^n)$ denote the size of the largest subset of $\mathbb{F}_p^n$ without a three-term arithmetic progression. Although similar methods have been used to study $\mathfrak{s}(G)$ and $r(\mathbb{F}_p^n)$, no direct connection between these quantities has previously been established. We give an upper bound for $\mathfrak{s}(G)$ in terms of $r(\mathbb{F}_p^n)$ for the prime divisors $p$ of $\operatorname{exp}(G)$. For the special case $G=\mathbb{F}_p^n$, we prove $\mathfrak{s}(\mathbb{F}_p^n)\leq 2p\cdot r(\mathbb{F}_p^n)$. Using the upper bounds for $r(\mathbb{F}_p^n)$ of Ellenberg and Gijswijt, this result improves the previously best known upper bounds for $\mathfrak{s}(\mathbb{F}_p^n)$ given by Naslund.
Eulerian Polynomials, Stirling Permutations of the Second Kind and Perfect Matchings
Authors: Ma Shi-Mei , Yeh Yeong-Nan
Published: January 2020

In this paper, we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles. Moreover, we get an expansion of the ordinary derangement polynomials in terms of the Stirling derangement polynomials. Finally, we present constructive proofs of a kind of combinatorial expansions of the Eulerian polynomials of types $A$ and $B$.
Many Triangles with Few Edges
Authors: Kirsch Rachel , Radcliffe A. J.
Published: January 2020

Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with $n$ vertices and maximum degree at most $r$, where $n = a(r+1)+b$ and $0 \le b \le r$, $aK_{r+1}\cup K_b$ has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh, and Sudakov conjectured that $aK_{r+1}\cup K_b$ also maximizes the number of complete subgraphs $K_t$ for each fixed size $t \ge 3$, and proved this for $a = 1$. Cutler and Radcliffe proved this conjecture for $r \le 6$. We investigate a variant of this problem where we fix the number of edges instead of the number of vertices. We prove that $aK_{r+1}\cup {\mathcal C}(b)$, where ${\mathcal C}(b)$ is the colex graph on $b$ edges, maximizes the number of triangles among graphs with $m$ edges and any fixed maximum degree $r\le 8$, where $m = a \binom{r+1}{2} + b$ and $0 \le b < \binom{r+1}{2}$.
Vizing's 2-Factor Conjecture Involving Toughness and Maximum Degree Conditions
Authors: Kanno Jinko , Shan Songling
Published: January 2020

Let $G$ be a simple graph, and let $\Delta(G)$ and $\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\chi'(G)=\Delta(G)$ or $\chi'(G)=\Delta(G)+1$. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $\Delta$-critical graph, then  $G$ has a 2-factor. Let $G$ be an $n$-vertex $\Delta$-critical graph. It was proved that if $\Delta(G)\ge n/2$, then $G$ has a 2-factor; and that if $\Delta(G)\ge 2n/3+13$, then $G$  has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a $\Delta$-critical graph under "moderate" given toughness and  maximum degree conditions. In particular, we show that  if $G$ is an  $n$-vertex $\Delta$-critical graph with toughness at least 3/2 and with maximum degree at least $n/3$, then $G$ has a 2-factor. We also construct a family of graphs that have order $n$, maximum degree $n-1$, toughness at least $3/2$, but have no 2-factor. This implies that the $\Delta$-criticality in the result is needed. In addition, we develop new techniques in proving the existence of 2-factors in graphs.
On Bergeron's Positivity Problem for $q$-Binomial Coefficients
Authors: Zanello Fabrizio
Published: January 2020

F. Bergeron recently asked the intriguing question whether $\binom{b+c}{b}_q -\binom{a+d}{d}_q$  has nonnegative coefficients as a polynomial in $q$, whenever $a,b,c,d$ are positive integers, $a$ is the smallest, and $ad=bc$. We conjecture that, in fact, this polynomial is also always unimodal, and combinatorially show our conjecture for $a\le 3$ and any $b,c\ge 4$. The main ingredient will be a novel (and rather technical) application of Zeilberger's KOH theorem.
Likelihood Orders for the $p$-Cycle Walks on the Symmetric Group
Authors: Bernstein Megan
Published: January 2020

Consider for a random walk on a group, the order from most to least likely element of the walk at each step, called the likelihood order. Up to periodicity issues, this order stabilizes after a sufficient number of steps. Here discrete Fourier analysis and the representations of the symmetric group, particularly formulas for the characters, are used to find the order after sufficient time for the random walks on the symmetric group generated by $p$-cycles for any $p$ fixed, $n$ sufficiently large. For the transposition walk, generated by all the $2$-cycles, at various levels of laziness, it is shown that order $n^2$ steps suffice for the order to stabilize.  Likelihood orders can aid in finding the total variation or separation distance mixing times.
Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences
Authors: Beaton Nicholas R. , Bouvel Mathilde , Guerrini Veronica , Rinaldi Simone
Published: January 2020

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.
Continuants, Run Lengths, and Barry's Modified Pascal Triangle
Authors: Spiegelhofer Lukas , Shallit Jeffrey
Published: January 2020

We show that the $n$'th diagonal sum of Barry's modified Pascal triangle can be described as the continuant of the run lengths of the binary representation of $n$.  We also obtain an explicit description for the row sums.
A Generalization of Erdős' Matching Conjecture
Authors: Pelekis Christos , Rocha Israel
Published: January 2020

Let $\mathcal{H}=(V,\mathcal{E})$ be an $r$-uniform hypergraph on $n$ vertices and fix a positive integer $k$ such that $1\le k\le r$. A $k$-matching of $\mathcal{H}$ is a collection of edges $\mathcal{M}\subset \mathcal{E}$ such that every subset of $V$ whose cardinality equals $k$ is contained in at most one element of $\mathcal{M}$. The $k$-matching number of $\mathcal{H}$ is the maximum cardinality of a $k$-matching. A well-known problem, posed by Erdős, asks for the maximum number of edges in an $r$-uniform hypergraph under constraints on its $1$-matching number. In this article we investigate the more general problem of determining the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices subject to the constraint that its $k$-matching number is strictly less than $a$. The problem can also be seen as a generalization of the well-known $k$-intersection problem. We propose candidate hypergraphs for the solution of this problem, and show that the extremal hypergraph is among this candidate set when $n\ge 4r\binom{r}{k}^2\cdot a$.
Antichain Toggling and Rowmotion
Authors: Joseph Michael
Published: January 2020

In this paper, we analyze the toggle group on the set of antichains of a poset. Toggle groups, generated by simple involutions, were first introduced by Cameron and Fon-Der-Flaass for order ideals of posets. Recently Striker has motivated the study of toggle groups on general families of subsets, including antichains. This paper expands on this work by examining the relationship between the toggle groups of antichains and order ideals, constructing an explicit isomorphism between the two groups (for a finite poset). We also focus on the rowmotion action on antichains of a poset that has been well-studied in dynamical algebraic combinatorics, describing it as the composition of antichain toggles. We also describe a piecewise-linear analogue of toggling to the Stanley’s chain polytope. We examine the connections with the piecewise-linear toggling Einstein and Propp introduced for order polytopes and prove that almost all of our results for antichain toggles extend to the piecewise-linear setting.
Vertex Covering with Monochromatic Pieces of few Colours
Authors: Eugster Marlo , Mousset Frank
Published: January 2020

In 1995, Erdös and Gyárfás proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2\sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers $r,s$, what is the smallest number $pc_{r,s}(K_n)$ such that in every colouring of the edges of $K_n$ with $r$ colours, there exists a vertex cover of $K_n$ by $pc_{r,s}(K_n)$ monochromatic paths using altogether at most $s$ different colours?For fixed integers $r>s$ and as $n\to\infty$, we prove that $pc_{r,s}(K_n) = \Theta(n^{1/\chi})$, where $\chi=\max{\{1,2+2s-r\}}$ is the chromatic number of the Kneser graph $KG(r,r-s)$. More generally, if one replaces $K_n$ by an arbitrary $n$-vertex graph with fixed independence number $\alpha$, then we have $pc_{r,s}(G) = O(n^{1/\chi})$, where this time around $\chi$ is the chromatic number of the Kneser hypergraph $KG^{(\alpha+1)}(r,r-s)$. This result is tight in the sense that there exist graphs with independence number $\alpha$ for which $pc_{r,s}(G) = \Omega(n^{1/\chi})$. This is in sharp contrast to the case $r=s$, where it follows from a result of Sárközy (2012) that $pc_{r,r}(G)$ depends only on $r$ and $\alpha$, but not on the number of vertices.We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic $d$-regular graphs.
On the Schur Positivity of $\Delta_{e_2} e_n[X]$
Authors: Dun Qiu , Remmel Jeffrey B. , Sergel Emily , Xin Guoce
Published: January 2020

Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson, and the second author have conjectured that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_k} e_n[X]$ is a polynomial in $\mathbb{N}[q,t]$. We present four proofs of a stronger statement in the case $k=2$; We show that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_2} e_n[X]$ has a positive expansion in terms of $q,t$-analogs.
Semiregular Automorphisms in Vertex-Transitive Graphs of Order $3p^2$
Authors: Marušič Dragan
Published: January 2020

It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally $2$-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. It is the purpose of this paper to prove that vertex-transitive graphs of order $3p^2$, where $p$ is a prime, contain semiregular automorphisms.
Uniquely $K_r^{(k)}$-Saturated Hypergraphs
Authors: Gyárfás András , Hartke Stephen G. , Viss Charles
Published: January 2020

In this paper we generalize the concept of uniquely $K_r$-saturated graphs to hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$ vertices. For integers $k,r,n$ such that $2\leqslant k <r<n$, a $k$-uniform hypergraph $H$ with $n$ vertices is uniquely $K_r^{(k)}$-saturated if $H$ does not contain $K_r^{(k)}$ but adding to $H$ any $k$-set that is not a hyperedge of $H$ results in exactly one copy of $K_r^{(k)}$. Among uniquely $K_r^{(k)}$-saturated hypergraphs, the interesting ones are the primitive ones that do not have a dominating vertex—a vertex belonging to all possible ${n-1\choose k-1}$ edges. Translating the concept to the complements of these hypergraphs, we obtain a natural restriction of $\tau$-critical hypergraphs: a hypergraph $H$ is uniquely $\tau$-critical if for every edge $e$, $\tau(H-e)=\tau(H)-1$ and $H-e$ has a unique transversal of size $\tau(H)-1$.We have two constructions for primitive uniquely $K_r^{(k)}$-saturated hypergraphs. One shows that for $k$ and $r$ where $4\leqslant kr$. This is in contrast to the case $k=2$ and $r=3$ where only the Moore graphs of diameter two have this property. Our other construction keeps $n-r$ fixed; in this case we show that for any fixed $k\ge 2$ there can only be finitely many examples. We give a range for $n$ where these hypergraphs exist. For $n-r=1$ the range is completely determined: $k+1\leqslant n \leqslant {(k+2)^2\over 4}$. For larger values of $n-r$ the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.