bluuebunny/crossref_metadata_2025_split huggingface dataset:
Partial List Colouring of Certain Graphs
Authors: Janssen Jeannette , Mathew Rogers , Rajendraprasad Deepak 
Published: January 2020

The partial list colouring conjecture due to Albertson, Grossman, and Haas (2000) states that for every $s$-choosable graph $G$ and every assignment of lists of size $t$, $1 \leq t \leq s$, to the vertices of $G$ there is an induced subgraph of $G$ on at least $\frac{t|V(G)|}{s}$ vertices which can be properly coloured from these lists. In this paper, we show that the partial list colouring conjecture holds true for certain classes of graphs like claw-free graphs, graphs with chromatic number at least $\frac{|V(G)|-1}{2}$, chordless graphs, and series-parallel graphs.
Decomposing Labeled Interval Orders as Pairs of Permutations
Authors: Claesson Anders , Hannah Stuart A. 
Published: January 2020

We introduce ballot matrices, a signed combinatorial structure whose definition naturally follows from the generating function for labeled interval orders. A sign reversing involution on ballot matrices is defined. We show that matrices fixed under this involution are in bijection with labeled interval orders and that they decompose to a pair consisting of a permutation and an inversion table. To fully classify such pairs, results pertaining to the enumeration of permutations having a given set of ascent bottoms are given. This allows for a new formula for the number of labeled interval orders.
Restricted Frame Graphs and a Conjecture of Scott
Authors: Chalopin Jérémie , Esperet Louis , Li Zhentao , Ossona de Mendez Patrice 
Published: January 2020

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdős). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once.It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once. We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.
The Degree/Diameter Problem in Maximal Planar Bipartite graphs
Authors: Dalfó Cristina , Huemer Clemens , Salas Julián 
Published: January 2020

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.
Normally Regular Digraphs
Authors: Jørgensen Leif K 
Published: January 2020

A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$, a pair of non-adjacent vertices have $\mu$ common out-neighbours, a pair of vertices connected by an edge in one direction have $\lambda$ common out-neighbours and a pair of vertices connected by edges in both directions have $2\lambda-\mu$ common out-neighbours. We often assume that two vertices can not be connected in both directions. We prove that the adjacency matrix of a normally regular digraph is normal. A connected $k$-regular digraph with normal adjacency matrix is a normally regular digraph if and only if all eigenvalues other than $k$ are on one circle in the complex plane. We prove several non-existence results, structural characterizations, and constructions of normally regular digraphs. In many cases these graphs are Cayley graphs of abelian groups and the construction is then based on a generalization of difference sets.We also show connections to other combinatorial objects: strongly regular graphs, symmetric 2-designs and association schemes.
Heffter Arrays and Biembedding Graphs on Surfaces
Authors: Archdeacon Dan 
Published: January 2020

A Heffter array is an $m \times n$ matrix with nonzero entries from $\mathbb{Z}_{2mn+1}$ such that i) every row and column sum to 0, and ii) exactly one of each pair $\{x,-x\}$ of nonzero elements appears in the array. We construct some Heffter arrays. These arrays are used to build current graphs used in topological graph theory. In turn, the current graphs are used to embed the complete graph $K_{2mn+1}$ so that the faces can be 2-colored, called a biembedding. Under certain conditions each color class forms a cycle system. These generalize biembeddings of Steiner triple systems. We discuss some variations including Heffter arrays with empty cells, embeddings on nonorientable surfaces, complete multigraphs, and using integer arithmetic in place of modular arithmetic.
Characterisations of Elementary Pseudo-Caps and Good Eggs
Authors: Rottey Sara , Van de Voorde Geertrui 
Published: January 2020

In this note, we use the theory of Desarguesian spreads to investigate good eggs. Thas showed that an egg in $PG(4n-1,q)$, $q$ odd, with two good elements is elementary. By a short combinatorial argument, we show that a similar statement holds for large pseudo-caps, in odd and even characteristic. As a corollary, this improves and extends the result of Thas, Thas and Van Maldeghem (2006) where one needs at least $4$ good elements of an egg in even characteristic to obtain the same conclusion. We rephrase this corollary to obtain a characterisation of the generalised quadrangle $T_3(O)$ of Tits. Lavrauw (2005) characterises elementary eggs in odd characteristic as those good eggs containing a space that contains at least $5$ elements of the egg, but not the good element. We provide an adaptation of this characterisation for weak eggs in odd and even characteristic. As a corollary, we obtain a direct geometric proof for the theorem of Lavrauw.
Combinatorics of Tableau Inversions
Authors: Beagley Jonathan E. , Drube Paul 
Published: January 2020

A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along with precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableau that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Finally, we share some the results of a computer program developed to calculate tableaux inversions.
On the Largest Component of a Hyperbolic Model of Complex Networks
Authors: Bode Michel , Fountoulakis Nikolaos , Müller Tobias 
Published: January 2020

We consider a model for complex networks that was introduced by Krioukov et al. In this model, $N$ points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an edge if they are within a certain hyperbolic distance. The $N$ points are distributed according to a quasi-uniform distribution, which is a distorted version of the uniform distribution. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering.
The model is controlled by two parameters $\alpha$ and $\nu$ where, roughly speaking, $\alpha$ controls the exponent of the power-law and $\nu$ controls the average degree. The present paper focuses on the evolution of the component structure of the random graph. We show that (a) for $\alpha > 1$ and $\nu$ arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; (b) for $\alpha < 1$ and $\nu$ arbitrary with high probability there is a "giant" component of linear order, and (c) when $\alpha=1$ then there is a non-trivial phase transition for the existence of a linear-sized component in terms of $\nu$.
A corrigendum was added to this paper 29 Dec 2018.
On the Isomorphism Problem for Cayley Graphs of Abelian Groups whose Sylow Subgroups are Elementary Abelian or Cyclic
Authors: Dobson Ted 
Published: January 2020

We show that if certain arithmetic conditions hold, then the Cayley isomorphism problem for abelian groups, all of whose Sylow subgroups are elementary abelian or cyclic, reduces to the Cayley isomorphism problem for its Sylow subgroups. This yields a large number of results concerning the Cayley isomorphism problem, perhaps the most interesting of which is the following: if $p_1,\ldots, p_r$ are distinct primes satisfying certain arithmetic conditions, then two Cayley digraphs of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$, $a_i\le 5$, are isomorphic if and only if they are isomorphic by a group automorphism of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$. That is, that such groups are CI-groups with respect to digraphs.
Random-Player Maker-Breaker games
Authors: Krivelevich Michael , Kronenberg Gal 
Published: January 2020

In a $(1:b)$ Maker-Breaker game, one of the central questions is to find the maximal value of $b$ that allows Maker to win the game (that is, the critical bias $b^*$). Erdős conjectured that the critical bias for many Maker-Breaker games played on the edge set of $K_n$ is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, "Erdős Paradigm" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly. A random-player Maker-Breaker game is a two-player game, played the same as an ordinary (biased) Maker-Breaker game, except that one player plays according to his best strategy and claims one element in each round, while the other plays randomly and claims $b$ (or $m$) elements. In fact, for every (ordinary) Maker-Breaker game, there are two different random-player versions; the $(1:b)$ random-Breaker game and the $(m:1)$ random-Maker game. We analyze the random-player version of several classical Maker-Breaker games such as the Hamilton cycle game, the perfect-matching game and the $k$-vertex-connectivity game (played on the edge set of $K_n$). For each of these games we find or estimate the asymptotic values of the bias (either $b$ or $m$) that allow each player to typically win the game. In fact, we provide the "smart" player with an explicit winning strategy for the corresponding value of the bias.
Connectivity of some Algebraically Defined Digraphs
Authors: Kodess Aleksandr , Lazebnik Felix 
Published: January 2020

Let $p$ be a prime, $e$ a positive integer, $q = p^e$, and let $\mathbb{F}_q$ denote the finite field of $q$ elements. Let $f_i\colon\mathbb{F}_q^2\to\mathbb{F}_q$ be arbitrary functions, where $1\le i\le l$, $i$ and $l$ are integers. The digraph $D = D(q;\bf{f})$, where ${\bf f}=f_1,\dotso,f_l)\colon\mathbb{F}_q^2\to\mathbb{F}_q^l$, is defined as follows. The vertex set of $D$ is $\mathbb{F}_q^{l+1}$. There is an arc from a vertex ${\bf x} = (x_1,\dotso,x_{l+1})$ to a vertex ${\bf y} = (y_1,\dotso,y_{l+1})$ if $x_i + y_i = f_{i-1}(x_1,y_1)$ for all $i$, $2\le i \le l+1$. In this paper we study the strong connectivity of $D$ and completely describe its strong components. The digraphs $D$ are directed analogues of some algebraically defined graphs, which have been studied extensively and have many applications.
The Three Colour Hat Guessing Game on Cycle Graphs
Authors: Szczechla Witold 
Published: January 2020

We study a cooperative game in which each member of a team of N players, wearing coloured hats and situated at the vertices of the cycle graph with N vertices, is guessing their own hat colour merely on the basis of observing the hats worn by their two neighbours without exchanging the information. Each hat can have one of three colours. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colours. We prove that a winning strategy exists if and only if N is divisible by 3 or N = 4. This asymmetric game is an example of relational system using incomplete information about an unpredictable situation, where at least one participant has to act properly.
Digraph Representations of 2-closed Permutation Groups with a Normal Regular Cyclic Subgroup
Authors: Xu Jing 
Published: January 2020

In this paper, we classify 2-closed (in Wielandt's sense) permutation groups which contain a normal regular cyclic subgroup and prove that for each such group $G$, there exists a circulant $\Gamma$ such that $\mathrm{Aut} (\Gamma)=G$.
Absolute Differences Along Hamiltonian Paths
Authors: Monopoli Francesco 
Published: January 2020

We prove that if the vertices of a complete graph are labeled with the elements of an arithmetic progression, then for any given vertex there is a Hamiltonian path starting at this vertex such that the absolute values of the differences of consecutive vertices along the path are pairwise distinct. In another extreme case where the label set has small additive energy, we show that the graph actually possesses a Hamiltonian cycle with the property just mentioned. These results partially solve a conjecture by Z.-W. Sun.
A Generalization of Graham’s Conjecture
Authors: Guan Huanhuan , Yuan Pingzhi , Zeng Xiangneng 
Published: January 2020

Let $G$ be a finite Abelian group of order $|G|=n$, and let $S=g_1\cdot\ldots\cdot g_{n-1}$ be a sequence over $G$ such that all nonempty zero-sum subsequences of $S$ have the same length. In this paper, we completely determine the structure of these sequences.
New Duality Operator for Complex Circulant Matrices and a Conjecture of Ryser
Authors: Gallardo Luis H. 
Published: January 2020

We associate to any given circulant complex matrix $C$ another one $E(C)$ such that $E(E(C)) = C^{*}$ the transpose conjugate of $C.$ All circulant Hadamard matrices of order $4$ satisfy a condition $C_4$ on their eigenvalues, namely, the absolute value of the sum of all eigenvalues is bounded above by $2.$ We prove by a "descent" that uses our operator $E$ that the only circulant Hadamard matrices of order $n \geq 4$, that satisfy a condition $C_n$ that generalizes the condition $C_4$ and that consist of a list of $n/4$ inequalities for the absolute value of some sums of eigenvalues of $H$ are the known ones.
Growth Rates of Groups associated with Face 2-Coloured Triangulations and Directed Eulerian Digraphs on the Sphere
Authors: McCourt Thomas A. 
Published: January 2020

Let $\mathcal{G}$ be a properly face $2$-coloured (say black and white) piecewise-linear triangulation of the sphere with vertex set $V$. Consider the abelian group $\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\mathcal{A}_W\cong\mathcal{A}_B\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathcal{C}$ where $\mathcal{C}$ is a finite abelian group.The finite torsion subgroup $\mathcal{C}$ is referred to as the canonical group of the triangulation. Let $m_t$ be the maximal order of $\mathcal{C}$ over all properly face 2-coloured spherical triangulations with $t$ triangles of each colour. By relating such a triangulation to certain directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to the triangulation's canonical group we provide improved upper and lower bounds for $\lim \sup_{t\rightarrow\infty}(m_t)^{1/t}$.
Weakly Distance-Regular Digraphs of Valency Three, I
Authors: Yang Yuefeng , Lv Benjian , Wang Kaishun 
Published: January 2020

Suzuki (2004) classified thin weakly distance-regular digraphs and proposed the project to classify weakly distance-regular digraphs of valency 3. The case of girth 2 was classified by the third author (2004) under the assumption of the commutativity. In this paper, we continue this project and classify these digraphs with girth more than 2 and two types of arcs.
Avoiding Letter Patterns in Ternary Square-Free Words
Authors: Petrova Elena A. 
Published: January 2020

We consider special patterns of lengths 5 and 6 in a ternary alphabet. We show that some of them are unavoidable in square-free words and prove avoidability of the other ones. Proving the main results, we use Fibonacci words as codes of ternary words in some natural coding system and show that they can be decoded to square-free words avoiding the required patterns. Furthermore, we estimate the minimal local (critical) exponents of square-free words with such avoidance properties.
The Smith and Critical Groups of the Square Rook's Graph and its Complement
Authors: Ducey Joshua E. , Gerhard Jonathan , Watson Noah 
Published: January 2020

Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column. This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$. In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement. This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs. In doing so we verify a 1986 conjecture of Rushanan.
Regular Graphs are Antimagic
Authors: Bérczi Kristóf , Bernáth Attila , Vizer Máté 
Published: January 2020

An undirected simple graph $G=(V,E)$ is called antimagic if there exists an injective function $f:E\rightarrow\{1,\dots,|E|\}$ such that $\sum_{e\in E(u)} f(e)\neq\sum_{e\in E(v)} f(e)$ for any pair of different nodes $u,v\in V$. In this note we prove — with a slight modification of an argument of Cranston et al. — that $k$-regular graphs are antimagic for $k\ge 2$.

A corrigendum was added to this paper on May 2, 2019.
Connected Even Factors in the Square of Essentially 2-Edge-connected Graph
Authors: Ekstein Jan , Wu Baoyindureng , Xiong Liming 
Published: January 2020

An essentially $k$-edge connected graph $G$ is a connected graph such that deleting less than $k$ edges from $G$ cannot result in two nontrivial components. In this paper we prove that if an essentially 2-edge-connected graph $G$ satisfies that for any pair of leaves at distance 4 in $G$ there exists another leaf of $G$ that has distance 2 to one of them, then the square $G^2$ has a connected even factor with maximum degree at most 4. Moreover we show that, in general, the square of essentially 2-edge-connected graph does not contain a connected even factor with bounded maximum degree.
An Extension of MacMahon's Equidistribution Theorem to Ordered Multiset Partitions
Authors: Wilson Andrew Timothy 
Published: January 2020

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work, we prove a strengthening of MacMahon's theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. This generalization leads to a new extension of Macdonald polynomials for hook shapes. We use our main theorem to show that these polynomials are symmetric and we give their Schur expansion.

A corrigendum was added 17 September 2019.
Triangular Fully Packed Loop Configurations of Excess 2
Authors: Beil Sabine 
Published: January 2020

Triangular fully packed loop configurations (TFPLs) came up in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. To a TFPL is assigned a triple $(u,v;w)$ of $01$-words encoding its boundary conditions which must necessarily satisfy that $d(u)+d(v)\leq d(w)$, where $d(u)$ denotes the number of inversions in $u$. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers of FPLs having given link patterns. Later, Wieland drift — a map on TFPLs that is based on Wieland gyration — was defined. The main contribution of this article will be a linear expression for the number of TFPLs with boundary $(u,v;w)$ where $d(w)-d(u)-d(v)=2$ in terms of numbers of stable TFPLs, that is, TFPLs invariant under Wieland drift. This linear expression generalises already existing enumeration results for TFPLs with boundary $(u,v;w)$ where $d(w)-d(u)-d(v)=0,1$.
On Some Conjectures Concerning Critical Independent Sets of a Graph
Authors: Short Taylor 
Published: January 2020

Let $G$ be a simple graph with vertex set $V(G)$. A set $S\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. For $X\subseteq V(G)$, the difference of $X$ is $d(X) = |X|-|N(X)|$ and an independent set $A$ is critical if $d(A) = \max \{d(X): X\subseteq V(G) \text{ is an independent set}\}$ (possibly $A=\emptyset$). Let $\text{nucleus}(G)$ and $\text{diadem}(G)$ be the intersection and union, respectively, of all maximum size critical independent sets in $G$. In this paper, we will give two new characterizations of Konig-Egervary graphs involving $\text{nucleus}(G)$ and $\text{diadem}(G)$. We also prove a related lower bound for the independence number of a graph. This work answers several conjectures posed by Jarden, Levit, and Mandrescu.