bluuebunny/crossref_metadata_2025_split huggingface dataset:
Turán $H$-Densities for 3-Graphs
Authors: Falgas-Ravry Victor , Vaughan Emil R. 
Published: January 2020

Given an $r$-graph $H$ on $h$ vertices, and a family $\mathcal{F}$ of forbidden subgraphs, we define $\mathrm{ex}_{H}(n, \mathcal{F})$ to be the maximum number of induced copies of $H$ in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Then the Turán $H$-density of $\mathcal{F}$ is the limit\[\pi_{H}(\mathcal{F})= \lim_{n\rightarrow \infty}\mathrm{ex}_{H}(n, \mathcal{F})/\binom{n}{h}. \]This generalises the notions of Turán density (when $H$ is an $r$-edge), and inducibility (when $\mathcal{F}$ is empty). Although problems of this kind have received some attention, very few results are known.We use Razborov's semi-definite method to investigate Turán $H$-densities for $3$-graphs. In particular, we show that\[\pi_{K_4^-}(K_4) = 16/27,\]with Turán's construction being optimal. We prove a result in a similar flavour for $K_5$ and make a general conjecture on the value of $\pi_{K_t^-}(K_t)$. We also establish that\[\pi_{4.2}(\emptyset)=3/4,\]where $4.2$ denotes the $3$-graph on $4$ vertices with exactly $2$ edges. The lower bound in this case comes from a random geometric construction strikingly different from previous known extremal examples in $3$-graph theory. We give a number of other results and conjectures for $3$-graphs, and in addition consider the inducibility of certain directed graphs. Let $\vec{S}_k$ be the out-star on $k$ vertices; i.e. the star on $k$ vertices with all $k-1$ edges oriented away from the centre. We show that\[\pi_{\vec{S}_3}(\emptyset)=2\sqrt{3}-3,\]with an iterated blow-up construction being extremal. This is related to a conjecture of Mubayi and Rödl on the Turán density of the 3-graph $C_5$. We also determine $\pi_{\vec{S}_k}(\emptyset)$ when $k=4,5$, and conjecture its value for general $k$.
On Graph Parameters Guaranteeing Fast Sandpile Diffusion
Authors: Choure Ayush , Vishwanathan Sundar 
Published: January 2020

The Abelian Sandpile Model (Dhar 1990) is a discrete diffusion process, defined on graphs, which serves as the standard model of self-organized criticality. One is allowed to add sand particles on the nodes of the graph such that each node can stably hold at most some bounded number of particles. The particles flow through the graph as a consequence of surpassing the node capacities, until they reach a special sink node possessing infinite capacity. These simple dynamics give rise to a very interesting Markovian system. The transience class of a sandpile is defined as the maximum number of particles that can be added without making the system recurrent. We identify a small set of key graph properties that guarantee polynomial bounds on transience classes of the sandpile families satisfying them. These properties governing the speed of sandpile diffusion process are volume growth parameters, boundary regularity type properties and non-empty interior type constraints.This generalizes a previous result by Babai and Gorodezky (2007), in which they establish polynomial bounds on the $n\times n$ grid. Indeed the properties we show are based on ideas extracted from their proof as well as the continuous analogs in the theory of harmonic functions.
A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas
Authors: Zaslavsky Thomas 
Published: January 2020

A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics.Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types:Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs.Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups — or no group.Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — except (to some extent) when the author calls the weights "signs".Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.
On the Maximum Number of $k$-Hooks of Partitions of $n$
Authors: Fan Anna R.B. , Yang Harold R.L. , Yu Rebecca T. 
Published: January 2020

Let $\alpha_k(\lambda)$ denote the number of $k$-hooks in a partition $\lambda$ and let $b(n,k)$ be the maximum value of $\alpha_k(\lambda)$ among partitions of $n$. Amdeberhan posed a conjecture on the generating function of $b(n,1)$. We give a proof of this conjecture. In general, we obtain a formula that can be used to determine $b(n,k)$. This leads to a generating function formula for $b(n,k)$. We introduce the notion of nearly $k$-triangular partitions. We show that for any $n$, there is a nearly $k$-triangular partition which can be transformed into a partition of $n$ that attains the maximum number of $k$-hooks. The operations for the transformation enable us to compute the number $b(n,k)$.
Towards Random Uniform Sampling of Bipartite Graphs with given Degree Sequence
Authors: Miklós István , Erdős Péter L , Soukup Lajos 
Published: January 2020

In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on $n$ vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in $n$ in case of half-regular degree sequence. The novelty of our approach lies in the construction of the multicommodity flow in Sinclair's method.
Nonrepetitive Colourings of Planar Graphs with $O(\log n)$ Colours
Authors: Dujmović Vida , Joret Gwenaël , Frati Fabrizio , Wood David R. 
Published: January 2020

A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph $G$ is the minimum integer $k$ such that $G$ has a nonrepetitive $k$-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is $O(\sqrt{n})$ for $n$-vertex planar graphs. We prove a $O(\log n)$ upper bound.
Limits of Modified Higher $q,t$-Catalan Numbers
Authors: Lee Kyungyong , Li Li , Loehr Nicholas A 
Published: January 2020

The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was eventually proved that all these definitions are equivalent. In this paper, we study the similar situation for higher $q,t$-Catalan numbers, where the equivalence of the algebraic and combinatorial definitions is still conjectural. We compute the limits of several versions of the modified higher $q,t$-Catalan numbers and show that these limits equal the generating function for integer partitions. We also identify certain coefficients of the higher $q,t$-Catalan numbers as enumerating suitable integer partitions, and we make some conjectures on the homological significance of the Bergeron-Garsia nabla operator.
Forbidden Triples Generating a Finite set of 3-Connected Graphs
Authors: Egawa Yoshimi , Fujisawa Jun , Furuya Michitaka , Plummer Michael D , Saito Akira 
Published: January 2020

For a graph $G$ and a set $\mathcal{F}$ of connected graphs, $G$ is said be $\mathcal{F}$-free if $G$ does not contain any member of $\mathcal{F}$ as an induced subgraph. We let $\mathcal{G} _{3}(\mathcal{F})$ denote the set of all $3$-connected $\mathcal{F}$-free graphs. This paper is concerned with sets $\mathcal{F}$ of connected graphs such that $|\mathcal{F}|=3$ and $\mathcal{G} _{3}(\mathcal{F})$ is finite. Among other results, we show that for an integer $m\geq 3$ and a connected graph $T$ of order greater than or equal to $4$, $\mathcal{G} _{3}(\{K_{4},K_{2,m},T\})$ is finite if and only if $T$ is a path of order $4$ or $5$.
Enumerating Finite Racks, Quandles and Kei
Authors: Blackburn Simon R. 
Published: January 2020

A rack of order $n$ is a binary operation $\vartriangleright$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\vartriangleright)$ is a rack provided that the map $x\mapsto x\vartriangleright y$ is a bijection for all $y\in X$, and $(x\vartriangleright y)\vartriangleright z=(x\vartriangleright z)\vartriangleright (y\vartriangleright z)$ for all $x,y,z\in X$.The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arbitrary rack is related to its operator group (the permutation group on $X$ generated by the maps $x\mapsto x\vartriangleright y$ for $y\in Y$), and then applying some of the theory of permutation groups. The relationship between a rack and its operator group extends results of Joyce and of Ryder; this relationship might be of independent interest.
Embedding Cycles in Finite Planes
Authors: Lazebnik Felix , Mellinger Keith E. , Vega Oscar 
Published: January 2020

We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$, a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective planes: for all $k$, $3\le k\le q^2+q+1$, a $k$-cycle can be embedded in any projective plane of order $q$.
A Criterion for the Log-Convexity of Combinatorial Sequences
Authors: Xia Ernest X.W. , Yao Olivia X.M. 
Published: January 2020

Recently, Došlić, and Liu and Wang developed techniques for dealing with the log-convexity of sequences. In this paper, we present a criterion for the log-convexity of some combinatorial sequences. In order to prove the log-convexity of a sequence satisfying a three-term recurrence, by our method, it suffices to compute a constant number of terms at the beginning of the sequence. For example, in order to prove the log-convexity of the Apéry numbers $A_n$, by our method, we just need to evaluate the values of $A_n$ for $0\leq n \leq 6$. As applications, we prove the log-convexity of some famous sequences including the Catalan-Larcombe-French numbers. This confirms a conjecture given by Sun.
Counting Words with Laguerre Series
Authors: Taylor Jair 
Published: January 2020

We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials with parameter $\alpha = -1$. We describe how such a series can be computed by finding an appropriate ordinary generating function and applying a certain transformation. We use this technique to find the generating function for the number of $k$-ary words avoiding any vincular pattern that has only ones, as well as words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length.
Near-Colorings: Non-Colorable Graphs and NP-Completeness
Authors: Montassier M. , Ochem P. 
Published: January 2020

A graph $G$ is $(d_1,...,d_l)$-colorable if the vertex set of $G$ can be partitioned into subsets $V_1,\ldots ,V_l$ such that the graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1 \leq i \leq l$. In this paper, we focus on complexity aspects of such colorings when $l=2,3$. More precisely, we prove that, for any fixed integers $k,j,g$ with $(k,j) \neq (0,0)$ and $g\geq3$, either every planar graph with girth at least $g$ is $(k,j)$-colorable or it is NP-complete to determine whether a planar graph with girth at least $g$ is $(k,j)$-colorable. Also, for any fixed integer $k$, it is NP-complete to determine whether a planar graph that is either $(0,0,0)$-colorable or non-$(k,k,1)$-colorable is $(0,0,0)$-colorable. Additionally, we exhibit non-$(3,1)$-colorable planar graphs with girth 5 and non-$(2,0)$-colorable planar graphs with girth 7.
A De Bruijn–Erdős Theorem for Chordal Graphs
Authors: Beaudou Laurent , Bondy Adrian , Chen Xiaomin , Chiniforooshan Ehsan , Chudnovsky Maria , Chvátal Vašek , Fraiman Nicolas , Zwols Yori 
Published: January 2020

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chávtal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs.
Spectra of Edge-Independent Random Graphs
Authors: Lu Linyuan , Peng Xing 
Published: January 2020

Let $G$ be a random graph on the vertex set $\{1,2,\ldots, n\}$ such that edges in $G$ are determined by independent random indicator variables, while the probabilities $p_{ij}$ for $\{i,j\}$ being an edge in $G$ are not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of $G$ are recently studied by Oliveira and Chung-Radcliffe. Let $A$ be the adjacency matrix of $G$, $\bar A={\rm E}(A)$, and $\Delta$ be the maximum expected degree of $G$. Oliveira first proved that asymptotically almost surely $\|A-\bar A\|=O(\sqrt{\Delta \ln n})$ provided $\Delta\geq C \ln n$ for some constant $C$. Chung-Radcliffe improved the hidden constant in the error term using a new Chernoff-type inequality for random matrices. Here we prove that asymptotically almost surely $\|A-\bar A\|\leq (2+o(1))\sqrt{\Delta}$ with a slightly stronger condition $\Delta\gg \ln^4 n$. For the Laplacian matrix $L$ of $G$, Oliveira and Chung-Radcliffe proved similar results $\|L-\bar L\|=O(\sqrt{\ln n}/\sqrt{\delta})$ provided the minimum expected degree $\delta\geq C' \ln n$ for some constant $C'$; we also improve their results by removing the $\sqrt{\ln n}$ multiplicative factor from the error term under some mild conditions. Our results naturally apply to the classical Erdős–Rényi random graphs, random graphs with given expected degree sequences, and bond percolation of general graphs.
On Limits of Dense Packing of Equal Spheres in a Cube
Authors: Tatarevic Milos 
Published: January 2020

We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of $\lceil p^{3}/2\rceil$ spheres. For this family of packings, the previous best-known arrangements were usually derived from a ccp by omission of a certain number of spheres without changing the initial structure. In this paper, we show that better arrangements exist for all $n\leq\lceil p^{3}/2\rceil-2$. We introduce an optimization method to reveal improvements of these packings, and present many new improvements for $n\leq4629$.
A Schur-Like Basis of NSym Defined by a Pieri Rule
Authors: Campbell John Maxwell , Feldman Karen , Light Jennifer , Shuldiner Pavel , Xu Yan 
Published: January 2020

Recent research on the algebra of non-commutative symmetric functions and the dual algebra of quasi-symmetric functions has explored some natural analogues of the Schur basis of the algebra of symmetric functions. We introduce a new basis of the algebra of non-commutative symmetric functions using a right Pieri rule. The commutative image of an element of this basis indexed by a partition equals the element of the Schur basis indexed by the same partition and the commutative image is $0$ otherwise. We establish a rule for right-multiplying an arbitrary element of this basis by an arbitrary element of the ribbon basis, and a Murnaghan-Nakayama-like rule for this new basis. Elements of this new basis indexed by compositions of the form $(1^n, m, 1^r)$ are evaluated in terms of the complete homogeneous basis and the elementary basis.
A Quantitative Study of Pure Parallel Processes
Authors: Bodini O. , Genitrini A. , Peschanski F. 
Published: January 2020

In this paper, we study the interleaving – or pure merge – operator that most often characterizes parallelism in concurrency theory. This operator is a principal cause of the so-called combinatorial explosion that makes the analysis of process behaviours e.g. by model-checking, very hard – at least from the point of view of computational complexity. The originality of our approach is to study this combinatorial explosion phenomenon on average, relying on advanced analytic combinatorics techniques. We study various measures that contribute to a better understanding of the process behaviours represented as plane rooted trees: the number of runs (corresponding to the width of the trees), the expected total size of the trees as well as their overall shape. Two practical outcomes of our quantitative study are also presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random sampling of concurrent runs. These provide interesting responses to the combinatorial explosion problem.
On Embeddings of Circulant Graphs
Authors: Conder Marston , Grande Ricardo 
Published: January 2020

A circulant of order $n$ is a Cayley graph for the cyclic group $\mathbb{Z}_n$, and as such, admits a transitive action of $\mathbb{Z}_n$ on its vertices. This paper concerns 2-cell embeddings of connected circulants on closed orientable surfaces. Embeddings on the sphere (the planar case) were classified by Heuberger (2003), and by a theorem of Thomassen (1991), there are only finitely many vertex-transitive graphs with minimum genus $g$, for any given integer $g \ge 3$. Here we completely determine all connected circulants with minimum genus 1 or 2; this corrects and extends an attempted classification of all toroidal circulants by Costa, Strapasson, Alves and Carlos (2010).
Proof of the List Edge Coloring Conjecture for Complete Graphs of Prime Degree
Authors: Schauz Uwe 
Published: January 2020

We prove that the list-chromatic index and paintability index of $K_{p+1}$ is $p$, for all odd primes $p$. This implies that the List Edge Coloring Conjecture holds for complete graphs with less then 10 vertices. It also shows that there are arbitrarily big complete graphs for which the conjecture holds, even among the complete graphs of class 1. Our proof combines the Quantitative Combinatorial Nullstellensatz with the Paintability Nullstellensatz and a group action on symmetric Latin squares. It displays various ways of using different Nullstellensätze. We also obtain a partial proof of a version of Alon and Tarsi's Conjecture about even and odd Latin squares.
Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets
Authors: Grinberg Darij , Roby Tom 
Published: January 2020

We study a birational map associated to any finite poset $P$. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of $P$. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call "skeletal". Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and "grafting onto an antichain"; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion.
An Abstraction of Whitney's Broken Circuit Theorem
Authors: Dohmen Klaus , Trinks Martin 
Published: January 2020

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $\sum_{A\subseteq S} f(A)$ where $S$ is a finite set and $f$ is a mapping from the power set of $S$ into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the Möbius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical Möbius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).
Some Combinatorial Arrays Related to the Lotka-Volterra System
Authors: Callan David , Ma Shi-Mei , Mansour Toufik 
Published: January 2020

The purpose of this paper is to investigate several context-free grammars suggested by the Lotka-Volterra system. Some combinatorial arrays, involving the Stirling numbers of the second kind and Eulerian numbers, are generated by these context-free grammars. In particular, we present grammatical characterization of some statistics on cyclically ordered partitions.
The Polytope of $k$-Star Densities
Authors: Rauh Johannes 
Published: January 2020

This paper describes the polytope $\mathbf{P}_{k;N}$ of $i$-star counts, for all $i\le k$, for graphs on $N$ nodes. The vertices correspond to graphs that are regular or as regular as possible. For even $N$ the polytope is a cyclic polytope, and for odd $N$ the polytope is well-approximated by a cyclic polytope. As $N$ goes to infinity, $\mathbf{P}_{k;N}$ approaches the convex hull of the moment curve. The affine symmetry group of $\mathbf{P}_{k;N}$ contains just a single non-trivial element, which corresponds to forming the complement of a graph.The results generalize to the polytope $\mathbf{P}_{I;N}$ of $i$-star counts, for $i$ in some set $I$ of non-consecutive integers. In this case, $\mathbf{P}_{I;N}$ can still be approximated by a cyclic polytope, but it is usually not a cyclic polytope itself.Polytopes of subgraph statistics characterize corresponding exponential random graph models. The elongated shape of the $k$-star polytope gives a qualitative explanation of some of the degeneracies found in such random graph models.
On the Real-Rootedness of the Descent Polynomials of $(n-2)$-Stack Sortable Permutations
Authors: Zhang Philip B. 
Published: January 2020

Bóna conjectured that the descent polynomials on $(n-2)$-stack sortable permutations have only real zeros. Brändén proved this conjecture by establishing a more general result. In this paper, we give another proof of Brändén's result by using the theory of $s$-Eulerian polynomials recently developed by Savage and Visontai.