bluuebunny/crossref_metadata_2025 huggingface dataset:
The Total Acquisition Number of Random Geometric Graphs
Authors: Infeld Ewa , Mitsche Dieter , Prałat Paweł
Published: January 2020

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process. In this paper, we investigate random geometric graphs $\mathcal{G}(n,r)$ with $n$ vertices distributed uniformally at random in $[0,\sqrt{n}]^2$ and two vertices being adjacent if and only if their distance is at most $r$. We show that asymptotically almost surely $a_t(\mathcal{G}(n,r)) = \Theta( n / (r \lg r)^2)$ for the whole range of $r=r_n \ge 1$ such that $r \lg r \le \sqrt{n}$. By monotonicity, asymptotically almost surely $a_t(\mathcal{G}(n,r)) = \Theta(n)$ if $r < 1$, and $a_t(\mathcal{G}(n,r)) = \Theta(1)$ if $r \lg r > \sqrt{n}$.
A Short Approach to Catalan Numbers Modulo $2^r$
Authors: Xin Guoce , Xu Jing-Feng
Published: January 2020

We notice that two combinatorial interpretations of the well-known Catalan numbers $C_n=(2n)!/n!(n+1)!$ naturally give rise to a recursion for $C_n$. This recursion is ideal for the study of the congruences of $C_n$ modulo $2^r$, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh's recent classification of $C_n$ modulo $2^r$. The equivalence $C_{n}\equiv_{2^r} C_{\bar n}$ is further reduced to $C_{n}\equiv_{2^r} C_{\tilde{n}}$ for simpler $\tilde{n}$. Moreover, by using connections between weighted Dyck paths and Motzkin paths, we find new classes of combinatorial sequences whose $2$-adic order is equal to that of $C_n$, which is one less than the sum of the digits of the binary expansion of $n+1$.
On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks
Authors: Eu Sen-Peng , Fu Tung-Shan , Liang Yu-Chang , Wong Tsai-Lien
Published: January 2020

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.
A Structural Characterization for Certifying Robinsonian Matrices
Authors: Laurent Monique , Seminaroti Matteo , Tanigawa Shin-ichi
Published: January 2020

A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is Robinsonian precisely when the graph is a unit interval graph, so that Robinsonian matrices form a matrix analogue of the class of unit interval graphs. Here we provide a structural characterization for Robinsonian matrices in terms of forbidden substructures, extending the notion of  asteroidal triples to weighted graphs. This implies the known characterization of unit interval graphs and leads to an efficient algorithm for certifying that a matrix is not Robinsonian.
Do Triangle-Free Planar Graphs have Exponentially Many $3$-Colorings?
Authors: Dvořák Zdeněk , Sereni Jean-Sébastien
Published: January 2020

Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $\alpha$ such that whenever $G$ is a planar graph and $A$ is a subset of its edges whose deletion makes $G$ triangle-free, there exists a subset $A'$ of $A$ of size at least $\alpha|A|$ such that $G-(A\setminus A')$ is $3$-colorable. This equivalence allows us to study restricted situations, where we can prove the statement to be true.
Parabolic Double Cosets in Coxeter Groups
Authors: Billey Sara C. , Konvalinka Matjaž , Petersen T. Kyle , Slofstra William , Tenner Bridget E.
Published: January 2020

Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$, as well as their ordinary and double quotients $W / W_I$ and $W_I \backslash W / W_J$, appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets $w W_I$, for $I \subseteq S$, forms the Coxeter complex of $W$, and is well-studied. In this article we look at a less studied object: the set of all double cosets $W_I w W_J$ for $I, J \subseteq S$. Double cosets are not uniquely presented by triples $(I,w,J)$. We describe what we call the lex-minimal presentation, and prove that there exists a unique such object for each double coset. Lex-minimal presentations are then used to enumerate double cosets via a finite automaton depending on the Coxeter graph for $(W,S)$. As an example, we present a formula for the number of parabolic double cosets with a fixed minimal element when $W$ is the symmetric group $S_n$ (in this case, parabolic subgroups are also known as Young subgroups). Our formula is almost always linear time computable in $n$, and we show how it can be generalized to any Coxeter group with little additional work. We spell out formulas for all finite and affine Weyl groups in the case that $w$ is the identity element.
Linear Bound for Majority Colourings of Digraphs
Authors: Knox Fiachra , Šámal Robert
Published: January 2020

Given $\eta \in [0, 1]$, a colouring $C$ of $V(G)$ is an $\eta$-majority colouring if at most $\eta d^+(v)$ out-neighbours of $v$ have colour $C(v)$, for any $v \in V(G)$. We show that every digraph $G$ equipped with an assignment of lists $L$, each of size at least $k$, has a $2/k$-majority $L$-colouring. For even $k$ this is best possible, while for odd $k$ the constant $2/k$ cannot be replaced by any number less than $2/(k+1)$. This generalizes a result of Anholcer, Bosek and Grytczuk, who proved the cases $k=3$ and $k=4$ and claim a weaker result for general $k$.
Induced Subgraphs of Graphs with Large Chromatic Number IX: Rainbow Paths
Authors: Scott Alex , Seymour Paul
Published: January 2020

We prove that for all integers $\kappa, s\ge 0$ there exists $c$ with the following property. Let $G$ be a graph with clique number at most $\kappa$ and chromatic number more than $c$. Then for every vertex-colouring (not necessarily optimal) of $G$, some induced subgraph of $G$ is an $s$-vertex path, and all its vertices have different colours. This extends a recent result of Gyárfás and Sárközy (2016) who proved the same for graphs $G$ with $\kappa=2$ and girth at least five.
Hamiltonicity in Locally Finite Graphs: Two Extensions and a Counterexample
Authors: Heuer Karl
Published: January 2020

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing $K^4$ or $K_{2,3}$ as a minor is Hamiltonian if and only if it is $2$-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the $2$-contractible edges. The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist.
Combinatorial Reductions for the Stanley Depth of $I$ and $S/I$
Authors: Keller Mitchel T. , Young Stephen J.
Published: January 2020

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal $I$ and the Stanley depth of its compliment, $S/I$. Using these results we are able to prove that if $S$ is a polynomial ring with at most 5 indeterminates and $I$ is a square-free monomial ideal, then the Stanley depth of $S/I$ is strictly larger than the Stanley depth of $I$. Using a computer search, we are able to extend this strict inequality up to polynomial rings with at most 7 indeterminates. This partially answers questions asked by Propescu and Qureshi as well as Herzog.
A Bound on Partitioning Clusters
Authors: Kane Daniel , Tao Terence
Published: January 2020

Let $X$ be a finite collection of sets (or "clusters"). We consider the problem of counting the number of ways a cluster $A \in X$ can be partitioned into two disjoint clusters $A_1, A_2 \in X$, thus $A = A_1 \uplus A_2$ is the disjoint union of $A_1$ and $A_2$; this problem arises in the run time analysis of the ASTRAL algorithm in phylogenetic reconstruction. We obtain the bound$$ | \{ (A_1,A_2,A) \in X \times X \times X: A = A_1 \uplus A_2 \} | \leq |X|^{3/p} $$where $|X|$ denotes the cardinality of $X$, and $p := \log_3 \frac{27}{4} = 1.73814\dots$, so that $\frac{3}{p} = 1.72598\dots$. Furthermore, the exponent $p$ cannot be replaced by any larger quantity. This improves upon the trivial bound of $|X|^2$. The argument relies on establishing a one-dimensional convolution inequality that can be established by elementary calculus combined with some numerical verification.In a similar vein, we show that for any subset $A$ of a discrete cube $\{0,1\}^n$, the additive energy of $A$ (the number of quadruples $(a_1,a_2,a_3,a_4)$ in $A^4$ with $a_1+a_2=a_3+a_4$) is at most $|A|^{\log_2 6}$, and that this exponent is best possible.
q-Stirling Identities Revisited
Authors: Cai Yue , Ehrenborg Richard , Readdy Margaret
Published: January 2020

We give combinatorial proofs of q-Stirling identities using restricted growth words.  This includes a poset theoretic proof of Carlitz's identity, a new proof of the q-Frobenius identity of Garsia and Remmel and of Ehrenborg's Hankel q-Stirling determinantal identity.  We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version.
Measures of Edge-Uncolorability of Cubic Graphs
Authors: Fiol M. A. , Mazzuoccolo G. , Steffen E.
Published: January 2020

There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the $5$-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for $3$-edge-colorable cubic graphs, cubic graphs which are not $3$-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measuring how far apart a non $3$-edge-colorable graph is from being $3$-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such  measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures.
Large Cuts with Local Algorithms on Triangle-Free Graphs
Authors: Hirvonen Juho , Rybicki Joel , Schmid Stefan , Suomela Jukka
Published: January 2020

Let $G$ be a $d$-regular triangle-free graph with $m$ edges. We present an algorithm which finds a cut in $G$ with at least $(1/2 + 0.28125/\sqrt{d})m$ edges in expectation, improving upon Shearer's classic result. In particular, this implies that any $d$-regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of $G$.Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm: each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques. We show that for any fixed $d$, there exists a weighted neighbourhood graph $\mathcal{N}_d$ such that there is a one-to-one correspondence between heavy cuts of $\mathcal{N}_d$ and randomised local algorithms that find large cuts in any $d$-regular input graph. This turns out to be a useful tool for analysing the existence of cuts in $d$-regular graphs: we can compute the optimal cut of $\mathcal{N}_d$ to attain a lower bound on the maximum cut size of any $d$-regular triangle-free graph.
The Spectral Gap of Graphs Arising From Substring Reversals
Authors: Chung Fan , Tobin Josh
Published: January 2020

The substring reversal graph $R_n$ is the graph whose vertices are the permutations $S_n$, and where two permutations are adjacent if one is obtained from a substring reversal of the other. We determine the spectral gap of $R_n$, and show that its spectrum contains many integer values. Further we consider a family of graphs that generalize the prefix reversal (or pancake flipping) graph, and show that every graph in this family has adjacency spectral gap equal to one.
Betti Numbers of Monomial Ideals and Shifted Skew Shapes
Authors: Nagel Uwe , Reiner Victor
Published: January 2020

We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field.
Bandwidth of Graphs Resulting from the Edge Clique Covering Problem
Authors: Engel Konrad , Hanisch Sebastian
Published: January 2020

Let $n,k,b$ be integers with $1 \le k-1 \le b \le n$ and let $G_{n,k,b}$ be the graph whose vertices are the $k$-element subsets $X$ of $\{0,\dots,n\}$ with $\mathrm{max}(X)-\mathrm{min}(X) \le b$ and where two such vertices $X,Y$ are joined by an edge if $\mathrm{max}(X \cup Y) - \mathrm{min}(X \cup Y) \le b$. These graphs are generated by applying a transformation to maximal $k$-uniform hypergraphs of bandwidth $b$ that is used to reduce the (weak) edge clique covering problem to a vertex clique covering problem. The bandwidth of $G_{n,k,b}$ is thus the largest possible bandwidth of any transformed $k$-uniform hypergraph of bandwidth $b$. For $b\geq \frac{n+k-1}{2}$, the exact bandwidth of these graphs is determined. Moreover, the bandwidth is determined asymptotically for $b=o(n)$ and for $b$ growing linearly in $n$ with a factor $\beta \in (0,1]$, where for one case only bounds could be found. It is conjectured that the upper bound of this open case is the right asymptotic value.
Chordal and Sequentially Cohen-Macaulay Clutters
Authors: Woodroofe Russ
Published: January 2020

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal non-chordal clutters are also closely related to obstructions to shellability, and we give some general families of such obstructions, together with a classification by computation of all obstructions to shellability on 6 vertices.
Fully Packed Loop Configurations: Polynomiality and Nested Arches
Authors: Aigner Florian
Published: January 2020

This article proves a conjecture by Zuber about the enumeration of fully packed loops (FPLs). The conjecture states that the number of FPLs whose link pattern consists of two noncrossing matchings which are separated by $m$ nested arches is a polynomial function in $m$ of certain degree and with certain leading coefficient. Contrary to the approach of Caselli, Krattenthaler, Lass and Nadeau (who proved a partial result) we make use of the theory of wheel polynomials developed by Di Francesco, Fonseca and Zinn-Justin. We present a new basis for the vector space of wheel polynomials and a polynomiality theorem in a more general setting. This allows us to finish the proof of Zubers conjecture.
Context-Free Grammars for Several Polynomials Associated with Eulerian Polynomials
Authors: Ma Shi-Mei , Ma Jun , Yeh Yeong-Nan , Zhu Bao-Xuan
Published: January 2020

In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials.
Binomial Edge Ideals with Quadratic Gröbner Bases
Authors: Crupi Marilena , Rinaldo Giancarlo
Published: January 2020

We prove that a binomial edge ideal of a graph $G$ has a quadratic Gröbner basis with respect to some term order if and only if the graph $G$ is closed with respect to a given labelling of the vertices. We also state some criteria for the closedness of a graph $G$ that do not depend on the labelling of its vertex set.
Conserved Quantities of Q-systems from Dimer Integrable Systems
Authors: Vichitkunakorn Panupong
Published: January 2020

We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems in Goncharov-Kenyon 2013. The dynamic on the graph is an urban renewal together with shrinking all 2-valent vertices, while it is a cluster transformation on the weight. The graph is not necessary obtained from an integral polygon. We define the Hamiltonians of a weighted graph as partition functions of all weighted perfect matchings with a common homology class, then show that they are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. This reproves the results of Di Francesco-Kedem 2010 and Galashin-Pylyavskyy 2016 for the Q-systems of type A, and gives new results for that of type B. Similar to the results in Di Francesco-Kedem 2010, the conserved quantities for Q-systems of type B can also be written as partition functions of hard particles on a certain graph. For type A, we show that the conserved quantities Poisson commute under a nondegenerate Poisson bracket.
Strong Chromatic Index of Graphs With Maximum Degree Four
Authors: Huang Mingfang , Santana Michael , Yu Gexin
Published: January 2020

A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In 1985, Erdős and Nešetřil conjectured that every graph with maximum degree $\Delta$ has a strong edge-coloring using at most $\frac{5}{4}\Delta^2$ colors if $\Delta$ is even, and at most $\frac{5}{4}\Delta^2 - \frac{1}{2}\Delta + \frac{1}{4}$ if $\Delta$ is odd. Despite recent progress for large $\Delta$ by using an iterative probabilistic argument, the only nontrivial case of the conjecture that has been verified is when $\Delta = 3$, leaving the need for new approaches to verify the conjecture for any $\Delta\ge 4$. In this paper, we apply some ideas used in previous results to an upper bound of 21 for graphs with maximum degree 4, which improves a previous bound due to Cranston in 2006 and moves closer to the conjectured upper bound of 20.
On the Rank and Periodic Rank of Finite Dynamical Systems
Authors: Gadouleau Maximilien
Published: January 2020

A finite dynamical system is a function $f : A^n \to A^n$ where $A$ is a finite alphabet, used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. The rank of a finite dynamical system is the cardinality of its image; the periodic rank is the number of its periodic points. In this paper, we determine the maximum rank and the maximum periodic rank of a finite dynamical system with a given interaction graph over any non-Boolean alphabet. The rank and the maximum rank are both computable in polynomial time. We also obtain a similar result for Boolean finite dynamical systems (also known as Boolean networks) whose interaction graphs are contained in a given digraph. We then prove that the average rank is relatively close (as the size of the alphabet is large) to the maximum. The results mentioned above only deal with the parallel update schedule. We finally determine the maximum rank over all block-sequential update schedules and the supremum periodic rank over all complete update schedules.
Coboundary Categories and Local Rules
Authors: Westbury Bruce W.
Published: January 2020

First we develop the theory of local rules for coboundary categories. Then we describe the local rules in two main cases. First for the quantum groups in general and in the seminormal representations of the Hecke algebras. Then for crystals in general and specifically for crystals of minuscule representations. Finally we show how growth diagrams can be extended to construct the action of the cactus group on highest weight words.
Hamilton Cycles in Infinite Cubic Graphs
Authors: Pitz Max F.
Published: January 2020

Investigating a problem of B. Mohar, we show that every one-ended Hamiltonian cubic graph with end degree 3 contains a second Hamilton cycle. We also construct two examples showing that this result does not extend to give a third Hamilton cycle, nor that it extends to the two-ended case.
Asymptotic Behaviour of the Number of Eulerian Circuits
Authors: Isaev Mikhail
Published: January 2020

We determine the asymptotic behaviour of the number of Eulerian circuits in undirected simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). We also prove some new properties of the Laplacian matrix.
Forbidden Subposet Problems for Traces of Set Families
Authors: Gerbner Dániel , Patkós Balázs , Vizer Máté
Published: January 2020

In this paper we introduce a problem that bridges forbidden subposet and forbidden subconfiguration problems. The sets $F_1,F_2, \dots,F_{|P|}$ form a copy of a poset $P$, if there exists a bijection $i:P\rightarrow \{F_1,F_2, \dots,F_{|P|}\}$ such that for any $p,p'\in P$ the relation $p<_P p'$ implies $i(p)\subsetneq i(p')$. A family $\mathcal{F}$ of sets is $P$-free if it does not contain any copy of $P$. The trace of a family $\mathcal{F}$ on a set $X$ is $\mathcal{F}|_X:=\{F\cap X: F\in \mathcal{F}\}$.We introduce the following notions: $\mathcal{F}\subseteq 2^{[n]}$ is $l$-trace $P$-free if for any $l$-subset $L\subseteq [n]$, the family $\mathcal{F}|_L$ is $P$-free and $\mathcal{F}$ is trace $P$-free if it is $l$-trace $P$-free for all $l\le n$. As the first instances of these problems we determine the maximum size of trace $B$-free families, where $B$ is the butterfly poset on four elements $a,b,c,d$ with $a,b<c,d$ and determine the asymptotics of  the maximum size of $(n-i)$-trace $K_{r,s}$-free families for $i=1,2$. We also propose a generalization of the main conjecture of the area of forbidden subposet problems.
Equitable Partitions to Spanning Trees in a Graph
Authors: Fekete Zsolt , Szabó Jácint
Published: January 2020

In this paper we first prove that if the edge set of an undirected graph is the disjoint union of two of its spanning trees, then for every subset $P$ of edges there exists a spanning tree decomposition that cuts $P$ into two (almost) equal parts. The main result of the paper is a further extension of this claim: If the edge set of a graph is the disjoint union of two of its spanning trees, then for every stable set of vertices of size 3, there exists such a spanning tree decomposition that cuts the stars of these vertices into (almost) equal parts. This result fails for 4 instead of 3. The proofs are elementary.
Constant Terms of Near-Dyson Polynomials
Authors: Gordeev Alexey
Published: January 2020

We formulate and prove a formula for the constant term for a certain class of Laurent polynomials, which include the Dyson conjecture and its generalizations by Bressoud and Goulden. Our method is explicit Combinatorial Nullstellensatz.
Sort-Invariant Non-Messing-Up
Authors: Tenner Bridget Eileen
Published: January 2020

A poset has the non-messing-up property if it has two covering sets of disjoint saturated chains so that for any labeling of the poset, sorting the labels along one set of chains and then sorting the labels along the other set yields a linear extension of the poset. The linear extension yielded by thus twice sorting a labeled non-messing-up poset may be independent of which sort was performed first. Here we characterize such sort-invariant labelings for convex subposets of a cylinder. They are completely determined by avoidance of a particular subpattern: a diamond of four elements whose smallest two labels appear at opposite points.
Finding Balance: Split Graphs and Related Classes
Authors: Collins Karen L. , Trenk Ann N.
Published: January 2020

A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. A split graph is unbalanced if there exist two such partitions that are distinct. Cheng, Collins and Trenk (2016), discovered the following interesting counting fact: unlabeled, unbalanced split graphs on $n$ vertices can be placed into a bijection with all unlabeled split graphs on $n-1$ or fewer vertices. In this paper we translate these concepts and the theorem to different combinatorial settings: minimal set covers, bipartite graphs with a distinguished block and posets of height one.
Residual $q$-Fano Planes and Related Structures
Authors: Etzion Tuvi , Hooker Niv
Published: January 2020

One of the most intriguing problems for $q$-analogs of designs, is the existence question of an infinite family of $q$-Steiner systems that are not spreads. In particular the most interesting case is the existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$-Steiner systems were found for one set of parameters. In this paper, a definition for the $q$-analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the $q$-analog properties. The existence of a design with the parameters of the residual $q$-Steiner system in general and the residual $q$-Fano plane in particular are examined. We construct different residual $q$-Fano planes for all $q$, where $q$ is a prime power. The constructed structure is just one step from a construction of a $q$-Fano plane.
On the Minimum Size of an Identifying Code Over All Orientations of a Graph
Authors: Cohen Nathann , Havet Frédéric
Published: January 2020

If $G$ be a graph or a digraph, let $\mathrm{id}(G)$ be the minimum size of an identifying code of $G$ if one exists, and $\mathrm{id}(G)=+\infty$ otherwise. For a graph $G$, let $\mathrm{idor}(G)$ be the minimum of $\mathrm{id}(D)$ overall orientations $D$ of $G$. We give some lower and upper bounds on $\mathrm{idor}(G)$. In particular, we show that $\mathrm{idor}(G)\leqslant \frac{3}{2} \mathrm{id}(G)$ for every graph $G$. We also show that computing $\mathrm{idor}(G)$ is NP-hard, while deciding whether $\mathrm{idor}(G)\leqslant |V(G)|-k$ is polynomial-time solvable for every fixed integer $k$.
Equidistributions of Mahonian Statistics over Pattern Avoiding Permutations
Authors: Amini Nima
Published: January 2020

A Mahonian $d$-function is a Mahonian statistic that can be expressed as a linear combination of vincular pattern statistics of length at most $d$. Babson and Steingrímsson classified all Mahonian 3-functions up to trivial bijections and identified many of them with well-known Mahonian statistics in the literature. We prove a host of Mahonian 3-function equidistributions over pattern avoiding sets of permutations. Tools used include block decomposition, Dyck paths and generating functions.
Nonexistence of Certain Singly Even Self-Dual Codes with Minimal Shadow
Authors: Bouyuklieva Stefka , Harada Masaaki , Munemasa Akihiro
Published: January 2020

It is known that there is no extremal singly even self-dual $[n,n/2,d]$ code with minimal shadow for $(n,d)=(24m+2,4m+4)$, $(24m+4,4m+4)$, $(24m+6,4m+4)$, $(24m+10,4m+4)$ and $(24m+22,4m+6)$. In this paper, we study singly even self-dual codes with minimal shadow having minimum weight $d-2$ for these $(n,d)$. For $n=24m+2$, $24m+4$ and $24m+10$, we show that the weight enumerator of a singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow is uniquely determined and we also show that there is no singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow for $m \ge 155$, $m \ge 156$ and $m \ge 160$, respectively. We demonstrate that the weight enumerator of a singly even self-dual code with minimal shadow is not uniquely determined for parameters $[24m+6,12m+3,4m+2]$ and $[24m+22,12m+11,4m+4]$.
The Tu–Deng Conjecture holds Almost Surely
Authors: Spiegelhofer Lukas , Wallner Michael
Published: January 2020

The Tu–Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base $2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and$1\leqslant t<2^k-1$. Then\[\Bigl \lvert\Bigl\{(a,b)\in\bigl\{0,\ldots,2^k-2\bigr\}^2:a+b\equiv t\bmod 2^k-1, w(a)+w(b)<k\Bigr\}\Bigr \rvert\leqslant 2^{k-1}.\]We prove that the Tu–Deng Conjecture holds almost surely in the following sense: the proportion of $t\in[1,2^k-2]$ such that the above inequality holds approaches $1$ as $k\rightarrow\infty$.Moreover, we prove that the Tu–Deng Conjecture implies a conjecture due to T. W. Cusick concerning the sum of digits of $n$ and $n+t$.
Eulerian Numbers Associated with Arithmetical Progressions
Authors: Ramírez José L. , Villamarin Sergio N. , Villamizar Diego
Published: January 2020

In this paper, we give a combinatorial interpretation of the $r$-Whitney-Eulerian numbers by means of coloured signed permutations. This sequence is a generalization of the well-known Eulerian numbers and it is connected to $r$-Whitney numbers of the second kind. Using generating functions, we provide some combinatorial identities and the log-concavity property. Finally, we show some basic congruences involving the $r$-Whitney-Eulerian numbers.
On a Conjecture Regarding Permutations which Destroy Arithmetic Progressions
Authors: Sawhney Mehtaab , Stoner David
Published: January 2020

Hegarty conjectured for $n\neq 2, 3, 5, 7$ that  $\mathbb{Z}/n\mathbb{Z}$ has a permutation which destroys all arithmetic progressions mod $n$. For $n\ge n_0$, Hegarty and Martinsson demonstrated that $\mathbb{Z}/n\mathbb{Z}$ has a permutation destroying arithmetic progressions. However $n_0\approx 1.4\times 10^{14}$ and thus resolving the conjecture in full remained out of reach of any computational techniques. Using constructions modeled after those used by Elkies and Swaminathan for the case of $\mathbb{Z}/p\mathbb{Z}$ with $p$ being prime, this paper establishes the conjecture in full. Furthermore, our results are completely independent of the proof given by Hegarty and Martinsson.
Pentavalent Symmetric Graphs of Order $12p$
Authors: Guo Song-Tao , Zhou Jin-Xin , Feng Yan-Quan
Published: January 2020

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, a complete classification of connected pentavalent symmetric graphs of order $12p$ is given for each prime $p$. As a result, a connected pentavalent symmetric graph of order $12p$ exists if and only if $p=2$, $3$, $5$ or $11$, and up to isomorphism, there are only nine such graphs: one for each $p=2$, $3$ and $5$, and six for $p=11$.
Improved Bounds for the Graham-Pollak Problem for Hypergraphs
Authors: Leader Imre , Tan Ta Sheng
Published: January 2020

For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-graphs needed to partition the complete $r$-graph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. An easy construction shows that $f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$, and we write $c_r$ for the least number such that $f_r(n) \leq c_r (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$.It was known that $c_r < 1$ for each even $r \geq 4$, but this was not known for any odd value of $r$. In this short note, we prove that $c_{295}<1$. Our method also shows that $c_r \rightarrow 0$, answering another open problem.
A New Upper Bound for Cancellative Pairs
Authors: Janzer Barnabás
Published: January 2020

A pair $(\mathcal{A},\mathcal{B})$ of families of subsets of an $n$-element set is called cancellative if whenever $A,A'\in\mathcal{A}$ and $B\in\mathcal{B}$ satisfy $A\cup B=A'\cup B$, then $A=A'$, and whenever $A\in\mathcal{A}$ and $B,B'\in\mathcal{B}$ satisfy $A\cup B=A\cup B'$, then $B=B'$. It is known that there exist cancellative pairs with $|\mathcal{A}||\mathcal{B}|$ about $2.25^n$, whereas the best known upper bound on this quantity is $2.3264^n$. In this paper we improve this upper bound to $2.2682^n$. Our result also improves the best known upper bound for Simonyi's sandglass conjecture for set systems.
More Turán-Type Theorems for Triangles in Convex Point Sets
Authors: Aronov Boris , Dujmović Vida , Morin Pat , Ooms Aurélien , Xavier da Silveira Luı́s Fernando Schultz
Published: January 2020

We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations.  As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact.  This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.
Schur-Concavity for Avoidance of Increasing Subsequences in Block-Ascending Permutations
Authors: Chen Evan
Published: January 2020

For integers $a_1, \dots, a_n \ge 0$ and $k \ge 1$, let $\mathcal L_{k+2}(a_1,\dots, a_n)$ denote the set of permutations of $\{1, \dots, a_1+\dots+a_n\}$ whose descent set is contained in $\{a_1, a_1+a_2, \dots, a_1+\dots+a_{n-1}\}$, and which avoids the pattern $12\dots(k+2)$. We exhibit some bijections between such sets, most notably showing that $\# \mathcal L_{k+2} (a_1, \dots, a_n)$ is symmetric in the $a_i$ and is in fact Schur-concave. This generalizes a set of equivalences observed by Mei and Wang.
On Upper Transversals in 3-Uniform Hypergraphs
Authors: Henning Michael A. , Yeo Anders
Published: January 2020

A set $S$ of vertices in a hypergraph $H$ is a transversal if it has a nonempty intersection with every edge of $H$. The upper transversal number $\Upsilon(H)$ of $H$ is the maximum cardinality of a minimal transversal in $H$. We show that if $H$ is a connected $3$-uniform hypergraph of order $n$, then $\Upsilon(H) > 1.4855 \sqrt[3]{n} - 2$. For $n$ sufficiently large, we construct infinitely many connected $3$-uniform hypergraphs, $H$, of order~$n$ satisfying $\Upsilon(H) < 2.5199 \sqrt[3]{n}$. We conjecture that $\displaystyle{\sup_{n \to \infty}  \, \left( \inf  \frac{ \Upsilon(H) }{ \sqrt[3]{n} } \right) = \sqrt[3]{16} }$, where the infimum is taken over all connected $3$-uniform hypergraphs $H$ of order $n$.
On Rainbow Hamilton Cycles in Random Hypergraphs
Authors: Dudek Andrzej , English Sean , Frieze Alan
Published: January 2020

Let $H_{n,p,r}^{(k)}$ denote a randomly colored random hypergraph, constructed on the vertex set $[n]$ by taking each $k$-tuple independently with probability $p$, and then independently coloring it with a random color from the set $[r]$. Let $H$ be a $k$-uniform hypergraph of order $n$. An $\ell$-Hamilton cycle is a spanning subhypergraph $C$ of $H$ with $n/(k-\ell)$ edges and such that for some cyclic ordering of the vertices each edge of $C$ consists of $k$ consecutive vertices and every pair of adjacent edges in $C$ intersects in precisely $\ell$ vertices.In this note we study the existence of rainbow $\ell$-Hamilton cycles (that is every edge receives a different color) in $H_{n,p,r}^{(k)}$. We mainly focus on the most restrictive case when $r = n/(k-\ell)$. In particular, we show that for the so called tight Hamilton cycles ($\ell=k-1$) $p = e^2/n$ is the sharp threshold for the existence of a rainbow tight Hamilton cycle in $H_{n,p,n}^{(k)}$ for each $k\ge 4$.
Antimagic Orientation of Biregular Bipartite Graphs
Authors: Shan Songling , Yu Xiaowei
Published: January 2020

An antimagic labeling of a directed graph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to the integers $\{1, \cdots, m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. An undirected graph $G$ is said to have an antimagic orientation if $G$ has an orientation which admits an antimagic labeling. Hefetz, Mütze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that every biregular bipartite graph admits an antimagic orientation.
Two-Vertex Generators of Jacobians of Graphs
Authors: Brandfonbrener David , Devlin Pat , Friedenberg Netanel , Ke Yuxuan , Marcus Steffen , Reichard Henry , Sciamma Ethan
Published: January 2020

We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We conclude with some conjectures about these divisors on random graphs and support them with empirical evidence.
Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box
Authors: Hartarsky Ivailo
Published: January 2020

In $r$-neighbour bootstrap percolation, vertices (sites) of a graph $G$ become "infected" in each round of the process if they have $r$ neighbours already infected. Once infected, they remain such. An initial set of infected sites is said to percolate if every site is eventually infected. We determine the maximal percolation time for $r$-neighbour bootstrap percolation on the hypercube for all $r \geq 3$ as the dimension $d$ goes to infinity up to a logarithmic factor. Surprisingly, it turns out to be $\frac{2^d}{d}$, which is in great contrast with the value for $r=2$, which is quadratic in $d$, as established by Przykucki (2012). Furthermore, we discover a link between this problem and a generalisation of the well-known Snake-in-the-Box problem.
Parallel Enumeration of Triangulations
Authors: Jordan Charles , Joswig Michael , Kastner Lars
Published: January 2020

We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing full triangulations only; this case is particularly relevant for tropical geometry. Most importantly, down-flip reverse search allows for massive parallelization, i.e., it scales well even for many cores. Our implementation allows to compute the triangulations of much larger point sets than before.
Bipartite Ramsey Numbers for Graphs of Small Bandwidth
Authors: Shen Lili , Lin Qizhong , Liu Qinghai
Published: January 2020

A graph $H=(W,E_H)$ is said to have bandwidth at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_H$, and a bipartite balanced $(\beta,\Delta)$-graph $H$ is a bipartite graph with bandwidth at most $\beta |W|$ and maximum degree at most $\Delta$, and furthermore it has a proper 2-coloring $\chi :W\rightarrow[2]$ such that $||\chi^{-1}(1)|-|\chi^{-1}(2)||\leq\beta|\chi^{-1}(2)|$. We prove that for any fixed $0<\gamma<1$ and integer $\Delta\ge1$, there exist a constant $\beta=\beta(\gamma,\Delta)>0$ and a natural number $n_0$ such that for every balanced $(\beta,\Delta)$-graph $H$ on $n\geq n_0$ vertices the bipartite Ramsey number $br(H,H)$ is at most $(1+\gamma)n$. In particular, $br(C_{2n},C_{2n})=(2+o(1))n$.
Inverse Perron Values and Connectivity of a Uniform Hypergraph
Authors: Bu Changjiang , Li Haifeng , Zhou Jiang
Published: January 2020

In this paper, we show that a uniform hypergraph $\mathcal{G}$ is connected if and only if one of its inverse Perron values is larger than $0$. We give some bounds on the bipartition width, isoperimetric number and eccentricities of $\mathcal{G}$ in terms of inverse Perron values. By using the inverse Perron values, we give an estimation of the edge connectivity of a $2$-design, and determine the explicit edge connectivity of a symmetric design. Moreover, relations between the inverse Perron values and resistance distance of a connected graph are presented.
Even Subgraph Expansions for the Flow Polynomial of Planar Graphs with Maximum Degree at Most 4
Authors: Deng Qingying , Jin Xian'an , Dong Fengming , Tay Eng Guan
Published: January 2020

As projections of links, 4-regular plane graphs are important in combinatorial knot theory. The flow polynomial of 4-regular plane graphs has a close relation with the two-variable Kauffman polynomial of links. F. Jaeger in 1991 provided even subgraph expansions for the flow polynomial of cubic plane graphs. Starting from and based on Jaeger's work, by introducing splitting systems of even subgraphs, we extend Jaeger's results from cubic plane graphs to plane graphs with maximum degree at most 4 including 4-regular plane graphs as special cases. Several consequences are derived and further work is discussed.
EW-Tableaux, Le-Tableaux, Tree-Like Tableaux and the Abelian Sandpile Model
Authors: Selig Thomas , Smith Jason P. , Steingrímsson Einar
Published: January 2020

A EW-tableau is a certain 0/1-filling of a Ferrers diagram, corresponding uniquely to an acyclic orientation, with a unique sink, of a certain bipartite graph called a Ferrers graph. We give a bijective proof of a result of Ehrenborg and van Willigenburg showing that EW-tableaux of a given shape are equinumerous with permutations with a given set of excedances. This leads to an explicit bijection between EW-tableaux and the much studied Le-tableaux, as well as the tree-like tableaux introduced by Aval, Boussicault and Nadeau.We show that the set of EW-tableaux on a given Ferrers diagram are in 1-1 correspondence with the minimal recurrent configurations of the Abelian sandpile model on the corresponding Ferrers graph.Another bijection between EW-tableaux and tree-like tableaux, via spanning trees on the corresponding Ferrers graphs, connects the tree-like tableaux to the minimal recurrent configurations of the Abelian sandpile model on these graphs. We introduce a variation on the EW-tableaux, which we call NEW-tableaux, and present bijections from these to Le-tableaux and tree-like tableaux. We also present results on various properties of and statistics on EW-tableaux and NEW-tableaux, as well as some open problems on these.
On the Real-Rootedness of the Local $h$-Polynomials of Edgewise Subdivisions
Authors: Zhang Philip B.
Published: January 2020

Athanasiadis conjectured that, for every positive integer $r$, the local $h$-polynomial of the $r$th edgewise subdivision of any simplex has only real zeros. In this paper, based on the theory of interlacing polynomials, we prove that a family of polynomials related to the desired local $h$-polynomial is interlacing and hence confirm Athanasiadis' conjecture.
Homotopy Type of the Neighborhood Complexes of Graphs of Maximal Degree at most $3$ and $4$-Regular Circulant Graphs
Authors: Shukla Samir
Published: January 2020

To estimate the lower bound for the chromatic number of a graph $G$,  Lovász associated a simplicial complex  $\mathcal{N}(G)$ called the neighborhood complex and related the topological connectivity of $\mathcal{N}(G)$ to the chromatic number of $G$. More generally he proved that the chromatic number of $G$ is bounded below by the topological connectivity of $\mathcal{N}(G)$ plus $3$. In this article, we consider the graphs of maximal degree at most $3$ and $4$-regular circulant graphs. We show that each connected component of the neighborhood complexes of these graphs is homotopy equivalent either to a point, to a wedge sum of circles, to a wedge sum of $2$-spheres $S^2$, to $S^3$, to a garland of $2$-spheres $S^2$ or to a connected sum of tori.
Crystals and Schur $P$-Positive Expansions
Authors: Choi Seung-Il , Kwon Jae-Hoon
Published: January 2020

We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $\mathfrak{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.
Counting Compositions over Finite Abelian Groups
Authors: Gao Zhicheng , MacFie Andrew , Wang Qiang
Published: January 2020

We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields can be converted to type (i) compositions, whereas Carlitz and "locally Mullen" compositions can be formulated as type (ii) compositions. We use the multisection formula to translate the problem from integers to group elements, the transfer matrix method to do exact counting, and finally the Perron-Frobenius theorem to derive asymptotics. We also exhibit bijections involving certain restricted classes of compositions.