bluuebunny/crossref_metadata_2025_split huggingface dataset:
Distant Set Distinguishing Total Colourings of Graphs
Authors: Przybyło Jakub 
Published: January 2020

The Total Colouring Conjecture suggests that $\Delta+3$ colours ought to suffice in order to provide a proper total colouring of every graph $G$ with maximum degree $\Delta$. Thus far this has been confirmed up to an additive constant factor, and the same holds even if one additionally requires every pair of neighbours in $G$ to differ with respect to the sets of their incident colours, so called pallets. Within this paper we conjecture that an upper bound of the form $\Delta+C$, for a constant $C>0$ still remains valid even after extending the distinction requirement to pallets associated with vertices at distance at most $r$, if only $G$ has minimum degree $\delta$ larger than a constant dependent on $r$. We prove that such assumption on $\delta$ is then unavoidable and exploit the probabilistic method in order to provide two supporting results for the conjecture. Namely, we prove the upper bound $(1+o(1))\Delta$ for every $r$, and show that for any fixed $\epsilon\in(0,1]$ and $r$, the conjecture holds if $\delta\geq \varepsilon\Delta$, i.e., in particular for regular graphs.
Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees
Authors: Blinovsky Vladimir , Greenhill Catherine 
Published: January 2020

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main result is an asymptotic enumeration formula for linear $r$-uniform hypergraphs with degree sequence $\boldsymbol{k}$. This formula holds whenever the maximum degree $k_{\max}$ satisfies $r^4 k_{\max}^4(k_{\max} + r) = o(M)$. Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.
Some Self-Orthogonal Codes Related to Higman's Geometry
Authors: Moori Jamshid , Rodrigues B. D. 
Published: January 2020

We examine some self-orthogonal codes constructed from a rank-5 primitive permutation representation of degree 1100 of the sporadic simple group ${\rm HS}$ of Higman-Sims. We show that ${\rm Aut}(C) = {\rm HS}{:}2$, where $C$ is a code of dimension 21 associated with Higman's geometry.
Words with many Palindrome Pair Factors
Authors: Borchert Adam , Rampersad Narad 
Published: January 2020

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many $n$, every length-$n$ factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue—Morse word does not have this property. We investigate finite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair.
Cycle Structures of Orthomorphisms Extending Partial Orthomorphisms of Boolean Groups
Authors: Schimanski Nichole L. , Caughman IV John S. 
Published: January 2020

A partial orthomorphism of a group $G$ (with additive notation) is an injection $\pi:S \to G$ for some $S \subseteq G$ such that $\pi(x)-x \not= \pi(y)-y$ for all distinct $x,y \in S$. We refer to $|S|$ as the size of $\pi$, and if $S = G$, then $\pi$ is an orthomorphism. Despite receiving a fair amount of attention in the research literature, many basic questions remain concerning the number of orthomorphisms of a given group, and what cycle types these permutations have.It is known that conjugation by automorphisms of $G$ forms a group action on the set of orthomorphisms of $G$. In this paper, we consider the additive group of binary $n$-tuples, $\mathbb{Z}_2^n$, where we extend this result to include conjugation by translations in $\mathbb{Z}_2^n$ and related compositions. We apply these results to show that, for any integer $n >1$, the distribution of cycle types of orthomorphisms of the group $\mathbb{Z}_2^n$ that extend any given partial orthomorphism of size two is independent of the particular partial orthomorphism considered. A similar result holds for size one. We also prove that the corresponding result does not hold for orthomorphisms extending partial orthomorphisms of size three, and we give a bound on the number of cycle-type distributions for the case of size three. As a consequence of these results, we find that all partial orthomorphisms of $\mathbb{Z}_2^n$ of size two can be extended to complete orthomorphisms.
Every Graph $G$ is Hall $\Delta(G)$-Extendible
Authors: Holliday Sarah , Vandenbussche Jennifer , Westlund Erik E. 
Published: January 2020

In the context of list coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list coloring. The graph $G$ with list assignment $L$, abbreviated $(G,L)$, satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $k$-extendible for some $k \geq \chi(G)$ if every $k$-precoloring of $G$ whose corresponding list assignment is Hall can be extended to a proper $k$-coloring of $G$. In 2011, Bobga et al. posed the question: If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-extendible? This paper establishes an affirmative answer to this question: every graph $G$ is Hall $\Delta(G)$-extendible. Results relating to the behavior of Hall extendibility under subgraph containment are also given. Finally, for certain graph families, the complete spectrum of values of $k$ for which they are Hall $k$-extendible is presented. We include a focus on graphs which are Hall $k$-extendible for all $k \geq \chi(G)$, since these are graphs for which satisfying the obviously necessary Hall's condition is also sufficient for a precoloring to be extendible.
Pathwidth and Nonrepetitive List Coloring
Authors: Gągol Adam , Joret Gwenaël , Kozik Jakub , Micek Piotr 
Published: January 2020

A vertex coloring of a graph is nonrepetitive if there is no path in the graph whose first half receives the same sequence of colors as the second half. While every tree can be nonrepetitively colored with a bounded number of colors (4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently showed that this does not extend to the list version of the problem, that is, for every $\ell \geq 1$ there is a tree that is not nonrepetitively $\ell$-choosable. In this paper we prove the following positive result, which complements the result of Fiorenzi et al.: There exists a function $f$ such that every tree of pathwidth $k$ is nonrepetitively $f(k)$-choosable. We also show that such a property is specific to trees by constructing a family of pathwidth-2 graphs that are not nonrepetitively $\ell$-choosable for any fixed $\ell$.
Two Partial Orders for Standard Young Tableaux
Authors: Kosakowska Justyna , Schmidmeier Markus , Thomas Hugh 
Published: January 2020

In this manuscript we show that two partial orders defined on the set of standard Young tableaux of shape $\alpha$ are equivalent. In fact, we give two proofs for the equivalence of the box order and the dominance order for tableaux. Both are algorithmic. The first of these proofs emphasizes links to the Bruhat order for the symmetric group and the second provides a more straightforward construction of the cover relations. This work is motivated by the known result that the equivalence of the two combinatorial orders leads to a description of the geometry of the representation space of invariant subspaces of nilpotent linear operators.
The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures
Authors: Braunfeld Samuel 
Published: January 2020

In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2.
On Symmetries in Phylogenetic Trees
Authors: Fusy Éric 
Published: January 2020

Billey et al. [arXiv:1507.04976] have recently discovered a surprisingly simple formula for the number $a_n(\sigma)$ of leaf-labelled rooted non-embedded binary trees (also known as phylogenetic trees) with $n\geq 1$ leaves, fixed (for the relabelling action) by a given permutation $\sigma\in\frak{S}_n$. Denoting by $\lambda\vdash n$ the integer partition giving the sizes of the cycles of $\sigma$ in non-increasing order, they show by a guessing/checking approach that if $\lambda$ is a binary partition (it is known that $a_n(\sigma)=0$ otherwise), then$$a_n(\sigma)=\prod_{i=2}^{\ell(\lambda)}(2(\lambda_i+\cdots+\lambda_{\ell(\lambda)})-1),$$and they derive from it a formula and random generation procedure for tanglegrams (and more generally for tangled chains). Our main result is a combinatorial proof of the formula for $a_n(\sigma)$, which yields a simplification of the random sampler for tangled chains.
Majority Bootstrap Percolation on $G(n,p)$
Authors: Holmgren Cecilia , Juškevičius Tomas , Kettle Nathan 
Published: January 2020

Majority bootstrap percolation on a graph $G$ is an epidemic process defined in the following manner. Firstly, an initially infected set of vertices is selected. Then step by step the vertices that have at least half of its neighbours infected become infected. We say that percolation occurs if eventually all vertices in $G$ become infected.In this paper we provide sharp bounds for the critical size of the initially infected set in majority bootstrap percolation on the Erdős-Rényi random graph $G(n,p)$. This answers an open question by Janson, Luczak, Turova and Vallier (2012). Our results obtained for $p=c\log(n)/n$ are close to the results obtained by Balogh, Bollobás and Morris (2009) for majority bootstrap percolation on the hypercube. We conjecture that similar results will be true for all regular-like graphs with the same density and sufficiently strong expansion properties.
A Square Root Map on Sturmian Words
Authors: Peltomäki Jarkko , Whiteland Markus A. 
Published: January 2020

We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope $\alpha$, there exists exactly six minimal squares in its language (a minimal square does not have a square as a proper prefix). A Sturmian word $s$ of slope $\alpha$ can be written as a product of these six minimal squares: $s = X_1^2 X_2^2 X_3^2 \cdots$. The square root of $s$ is defined to be the word $\sqrt{s} = X_1 X_2 X_3 \cdots$. The main result of this paper is that $\sqrt{s}$ is also a Sturmian word of slope $\alpha$. Further, we characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of $\sqrt{s}$ and an occurrence of any prefix of $\sqrt{s}$ in $s$. Related to the square root map, we characterize the solutions of the word equation $X_1^2 X_2^2 \cdots X_n^2 = (X_1 X_2 \cdots X_n)^2$ in the language of Sturmian words of slope $\alpha$ where the words $X_i^2$ are minimal squares of slope $\alpha$.We also study the square root map in a more general setting. We explicitly construct an infinite set of non-Sturmian fixed points of the square root map. We show that the subshifts $\Omega$ generated by these words have a curious property: for all $w \in \Omega$ either $\sqrt{w} \in \Omega$ or $\sqrt{w}$ is periodic. In particular, the square root map can map an aperiodic word to a periodic word.
The Threshold for Jigsaw Percolation on Random Graphs
Authors: Bollobás Béla , Riordan Oliver , Slivken Erik , Smith Paul 
Published: January 2020

Jigsaw percolation is a model for the process of solving puzzles within a social network, which was recently proposed by Brummitt, Chatterjee, Dey and Sivakoff. In the model there are two graphs on a single vertex set (the `people' graph and the `puzzle' graph), and vertices merge to form components if they are joined by an edge of each graph. These components then merge to form larger components if again there is an edge of each graph joining them, and so on. Percolation is said to occur if the process terminates with a single component containing every vertex. In this note we determine the threshold for percolation up to a constant factor, in the case where both graphs are Erdős-Rényi random graphs.
Symmetries of Unlabelled Planar Triangulations
Authors: Kang Mihyun , Sprüssel Philipp 
Published: January 2020

We derive a decomposition scheme of unlabelled triangulations rooted at a single cell, where the decomposition depends on whether the automorphism group of the triangulation contains reflections, rotations, or both. Furthermore, the decomposition scheme is constructive in the sense that for each of the three cases, there is a $k\in\mathbb{N}$ such that the scheme defines a one-to-$k$ correspondence between the respective triangulations and their decompositions.
Matching and Independence Complexes Related to Small Grids
Authors: Braun Benjamin , Hough Wesley K. 
Published: January 2020

The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(\Delta_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(\Delta_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(\Delta_n^m)$ and prove that several homology groups of $\mathrm{Ind}(\Delta_n^m)$ are non-zero.
Excluding Hooks and their Complements
Authors: Choromanski Krzysztof , Falik Dvir , Liebenau Anita , Patel Viresh , Pilipczuk Marcin 
Published: January 2020

The long-standing Erdős-Hajnal conjecture states that for every $n$-vertex undirected graph $H$ there exists $\epsilon(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or an independent set of size at least $n^{\epsilon(H)}$. A natural weakening of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both $H$ and its complement $H^\mathrm{c}$. These conjectures have been shown to hold for only a handful of graphs: it is not even known if they hold for all graphs on $5$ vertices.In a recent breakthrough, the symmetrized version of the Erdős-Hajnal conjecture was shown to hold for all paths. The goal of this paper is to show that the symmetrized conjecture holds for all trees on $6$ (or fewer) vertices. In fact this is a consequence of showing that the symmetrized conjecture holds for any path with a pendant edge at its third vertex; thus we also give a new infinite family of graphs for which the symmetrized conjecture holds.
Neighborhood Growth Dynamics on the Hamming Plane
Authors: Gravner Janko , Sivakoff David , Slivken Erik 
Published: January 2020

We initiate the study of general neighborhood growth dynamics on two-dimensional Hamming graphs. The decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. We focus on two related extremal quantities. The first is the size of the smallest set that eventually occupies the entire plane. The second is the minimum of an energy-entropy functional that comes from the scaling of the probability of eventual full occupation versus the density of the initial product measure within a rectangle. We demonstrate the existence of this scaling and study these quantities for large Young diagrams.
A New Bijective Proof of Babson and Steingrímsson's Conjecture
Authors: Chen Joanna N. , Li Shouxiao 
Published: January 2020

Babson and Steingrímsson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $\pi$, let $des(\pi)$ denote the descent number of $\pi$ and $maj(\pi)$ denote the major index of $\pi$. Babson and Steingrímsson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.
On Robust Colorings of Hamming-Distance Graphs
Authors: Harney Isaiah , Gluesing-Luerssen Heide 
Published: January 2020

$H_q(n,d)$ is defined as the graph with vertex set $\mathbb{Z}_q^n$ and where two vertices are adjacent if their Hamming distance is at least $d$. The chromatic number of these graphs is presented for various sets of parameters $(q,n,d)$. For the $4$-colorings of the graphs $H_2(n,n-1)$ a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust $4$-colorings of $H_2(n,n-1)$ is presented.
Dinv and Area
Authors: Garsia Adriano , Xin Guoce 
Published: January 2020

We give a new combinatorial proof of the well known result that the dinv of an $(m,n)$-Dyck path is equal to the area of its sweep map image. The first proof of this remarkable identity for co-prime $(m,n)$ is due to Loehr and Warrington. There is also a second proof (in the co-prime case) due to Gorsky and Mazin and a third proof due to Mazin. A corrigendum was added to this paper on the 9th of June 2017.
Avoidability of Formulas with Two Variables
Authors: Ochem Pascal , Rosenfeld Matthieu 
Published: January 2020

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We consider the patterns such that at most two variables appear at least twice, or equivalently, the formulas with at most two variables. For each such formula, we determine whether it is $2$-avoidable, and if it is $2$-avoidable, we determine whether it is avoided by exponentially many binary words.
The Turán Polytope
Authors: Raymond Annie 
Published: January 2020

The Turán hypergraph problem asks to find the maximum number of $r$-edges in a $r$-uniform hypergraph on $n$ vertices that does not contain a clique of size $a$. When $r=2$, i.e., for graphs, the answer is well-known and can be found in Turán's theorem. However, when $r\ge 3$, the problem remains open. We model the problem as an integer program and call the underlying polytope the Turán polytope. We draw parallels between the latter and the stable set polytope: we show that generalized and transformed versions of the web and wheel inequalities are also facet-defining for the Turán polytope. We also show that clique inequalities and what we call doubling inequalities are facet-defining when $r=2$. These facets lead to a simple new polyhedral proof of Turán's theorem.
Algebraic Properties of Chromatic Roots
Authors: Cameron Peter J. , Morgan Kerri 
Published: January 2020

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.
$\delta$-Connectivity in Random Lifts of Graphs
Authors: Silas Shashwat 
Published: January 2020

Amit and Linial have shown that a random lift of a connected graph with minimum degree $\delta\ge3$ is asymptotically almost surely (a.a.s.) $\delta$-connected and mentioned the problem of estimating this probability as a function of the degree of the lift. Using a connection between a random $n$-lift of a graph and a randomly generated subgroup of the symmetric group on $n$-elements, we show that this probability is at least $1 - O\left(\frac{1}{n^{\gamma(\delta)}}\right)$ where $\gamma(\delta)>0$ for $\delta\ge 5$ and it is strictly increasing with $\delta$. We extend this to show that one may allow $\delta$ to grow slowly as a function of the degree of the lift and the number of vertices and still obtain that random lifts are a.a.s. $\delta$-connected. We also simplify a later result showing a lower bound on the edge expansion of random lifts. On a related note, we calculate the probability that a subgroup of a wreath product of symmetric groups generated by random generators is transitive, extending a well known result of Dixon which covers the case for subgroups of the symmetric group.