bluuebunny/crossref_metadata_2025 huggingface dataset:
Powerful Sets: a Generalisation of Binary Matroids
Authors: Farr Graham E. , Wang Andrew Y.Z.
Published: January 2020

A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a powerful code if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2.  By treating binary vectors as characteristic vectors of subsets of $E$, we say that a set $S\subseteq2^E$ of subsets of $E$ is a powerful set if the set of characteristic vectors of sets in $S$ is a powerful code. Powerful sets (codes) include cocircuit spaces of binary matroids (equivalently, linear codes over $\mathbb{F}_2$), but much more besides. Our motivation is that, to each powerful set, there is an associated nonnegative-integer-valued rank function (by a construction of Farr), although it does not in general satisfy all the matroid rank axioms.In this paper we investigate the combinatorial properties of powerful sets.  We prove fundamental results on special elements (loops, coloops, frames, near-frames, and stars), their associated types of single-element extensions, various ways of combining powerful sets to get new ones, and constructions of nonlinear powerful sets. We show that every powerful set is determined by its clutter of minimal nonzero members.  Finally, we show that the number of powerful sets is doubly exponential, and hence that almost all powerful sets are nonlinear.
Algebraically Grid-Like Graphs have Large Tree-Width
Authors: Weißauer Daniel
Published: January 2020

By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of 'grid-likeness' of a graph.

The grid contains a long cycle on the perimeter, which is the $\mathbb{F}_2$-sum of the rectangles inside. Moreover, the grid distorts the metric of the cycle only by a factor of two. We prove that every graph that resembles the grid in this algebraic sense has large tree-width:

Let $k, p$ be integers, $\gamma$ a real number and $G$ a graph. Suppose that $G$ contains a cycle of length at least $2 \gamma p k$ which is the $\mathbb{F}_2$-sum of cycles of length at most $p$ and whose metric is distorted by a factor of at most $\gamma$. Then $G$ has tree-width at least $k$.
On Cartesian Products which Determine Few Distinct Distances
Authors: Pohoata Cosmin
Published: January 2020

Every set of points $\mathcal{P}$ determines $\Omega(|\mathcal{P}| / \log |\mathcal{P}|)$ distances. A close version of this was initially conjectured by Erdős in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set $\mathcal{P}$ of the form $A \times A$ must satisfy $|A - A| \ll |A|^{2-\frac{2}{7}} \log^{\frac{1}{7}} |A|$. This improves recent results of Hanson and Roche-Newton.
On the Alon-Tarsi Number and Chromatic-Choosability of Cartesian Products of Graphs
Authors: Kaul Hemanshu , Mudrock Jeffrey A.
Published: January 2020

We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest $k$ for which there is an orientation, $D$, of $G$ with max indegree $k\!-\!1$ such that the number of even and odd circulations contained in $D$ are different. It is known that $\chi(G) \leq \chi_\ell(G) \leq \chi_p(G) \leq AT(G)$, where  $\chi(G)$ is the chromatic number, $\chi_\ell(G)$ is the list chromatic number, and $\chi_p(G)$ is the paint number of $G$. In this paper we find families of graphs $G$ and $H$ such that $\chi(G \square H) = AT(G \square H)$, reducing this sequence of inequalities to equality.

We show that the Alon-Tarsi number of the Cartesian product of an odd cycle and a path is always equal to 3. This result is then extended to show that if $G$ is an odd cycle or a complete graph and $H$ is a graph on at least two vertices containing the Hamilton path $w_1, w_2, \ldots, w_n$ such that for each $i$, $w_i$ has a most $k$ neighbors among $w_1, w_2, \ldots, w_{i-1}$, then $AT(G \square H) \leq \Delta(G)+k$ where $\Delta(G)$ is the maximum degree of $G$.  We discuss other extensions for $G \square H$, where $G$ is such that $V(G)$ can be partitioned into odd cycles and complete graphs, and $H$ is a graph containing a Hamiltonian path. We apply these bounds to get chromatic-choosable Cartesian products, in fact we show that these families of graphs have $\chi(G) = AT(G)$, improving previously known bounds.
Counting Proper Colourings in 4-Regular Graphs via the Potts Model
Authors: Davies Ewan
Published: January 2020

We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $q\ge 5$. This proves the first case of a conjecture of the author, Perkins, Jenssen and Roberts, and implies tight bounds on the antiferromagnetic Potts partition function. The zero-temperature limit gives upper and lower bounds on the number of proper $q$-colourings of $4$-regular graphs, which almost proves the case $d=4$ of a conjecture of Galvin and Tetali. For any $q \ge 5$ we prove that the number of proper $q$-colourings of a $4$-regular graph is maximised by a union of $K_{4,4}$'s.
Boxicity, Poset Dimension, and Excluded Minors
Authors: Esperet Louis , Wiechert Veit
Published: January 2020

In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs.
A Bijection between Necklaces and Multisets with Divisible Subset Sum
Authors: Chan Swee Hong
Published: January 2020

Consider these two distinct combinatorial objects: (1) the necklaces of length $n$ with at most $q$ colors, and (2) the multisets of integers modulo $n$ with subset sum divisible by $n$ and with the  multiplicity of each element being strictly less than $q$. We show that these two objects have the same cardinality if $q$ and $n$ are mutually coprime. Additionally, when  $q$ is a prime power, we construct a bijection between these two objects by viewing  necklaces as  cyclic polynomials over the finite field of size $q$. Specializing to $q=2$ answers a  bijective problem posed by Richard  Stanley (Enumerative Combinatorics Vol. 1 Chapter 1, Problem 105(b)).
The Diachromatic Number of Digraphs
Authors: Araujo-Pardo Gabriela , Montellano-Ballesteros Juan José , Olsen Mika , Rubio-Montiel Christian
Published: January 2020

We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromaticnumber is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichromatic and diachromatic numbers are, respectively, the smallest and the largest number of colors in a complete acyclic coloring. We give some general results for the diachromatic number and study it for tournaments. We also show that the interpolation property for complete acyclic colorings does hold and establish Nordhaus-Gaddum relations.
Ramsey Numbers of Ordered Graphs
Authors: Balko Martin , Cibulka Josef , Král Karel , Kynčl Jan
Published: January 2020

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices.
The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$.
In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number.
The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.
Bootstrap Percolation in Directed Inhomogeneous Random Graphs
Authors: Meyer-Brandis Thilo , Detering Nils , Panagiotou Konstantinos
Published: January 2020

Bootstrap percolation is a process that is used to describe the spread of an infection on a given graph. In the model considered here each vertex is equipped with an individual threshold. As soon as the number of infected neighbors exceeds that threshold, the vertex gets infected as well and remains so forever. We perform a thorough analysis of bootstrap percolation on a novel model of directed and inhomogeneous random graphs, where the distribution of the edges is specied by assigning two distinct weights to each vertex, describing the tendency of it to receive edges from or to send edges to other vertices. Under the mild assumption that the limiting degree distribution of the graph is integrable we determine the typical fraction of infected vertices. Our model allows us to study a variety of settings, in particular the prominent case in which the degree distribution has an unbounded variance. As a second main contribution, we quantify the notion of "systemic risk", that is, we characterize to what extent tiny initial infections can propagate to large parts of the graph through a cascade, and discover novel features that make graphs prone/resilient to initially small infections.
An Edmonds-Gallai-Type Decomposition for the j-Restricted k-Matching Problem
Authors: Li Yanjun , Szabó Jácint
Published: January 2020

Given a non-negative integer $j$ and a positive integer $k$, a $j$-restricted $k$-matching in a simple undirected graph is a $k$-matching, so that each of its connected components has at least $j+1$ edges. The maximum non-negative node weighted $j$-restricted $k$-matching problem was recently studied by Li who gave a polynomial-time algorithm and a min-max theorem for $0 \leqslant j < k$, and also proved the NP-hardness of the problem with unit node weights and $2 \leqslant k \leqslant j$. In this paper we derive an Edmonds–Gallai-type decomposition theorem for the $j$-restricted $k$-matching problem with $0 \leqslant j < k$, using the analogous decomposition for $k$-piece packings given by Janata, Loebl and Szabó, and give an alternative proof to the min-max theorem of Li.
On the Clique Number of a Strongly Regular Graph
Authors: Greaves Gary R. W. , Soicher Leonard H.
Published: January 2020

We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including infinitely many parameter tuples that correspond to Paley graphs.
Maximum Number of Colourings: 5-Chromatic Case
Authors: Knox Fiachra , Mohar Bojan
Published: January 2020

In 1971, Tomescu conjectured [Le nombre des graphes connexes $k$-chromatiques minimaux aux sommets étiquetés, C. R. Acad. Sci. Paris 273 (1971), 1124--1126] that every connected graph $G$ on $n$ vertices with $\chi(G) = k \geq 4$ has at most $k!(k-1)^{n-k}$ $k$-colourings, where equality holds if and only if the graph is formed from $K_k$ by repeatedly adding leaves. In this note we prove (a strengthening of) the conjecture of Tomescu when $k=5$.
Planar Transitive Graphs
Authors: Hamann Matthias
Published: January 2020

We prove that the first homology group of every planar locally finite transitive graph $G$ is finitely generated as an $\Aut(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible.
Decomposing 1-Sperner Hypergraphs
Authors: Boros Endre , Gurvich Vladimir , Milanič Martin
Published: January 2020

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of $1$-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of $1$-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that $1$-Sperner hypergraphs are both threshold and equilizable. The study of $1$-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper.
On Some Interesting Ternary Formulas
Authors: Ochem Pascal , Rosenfeld Matthieu
Published: January 2020

We obtain the following results about the avoidance of ternary formulas. Up to renaming of the letters, the only infinite ternary words avoiding the formula $ABCAB.ABCBA.ACB.BAC$ (resp. $ABCA.BCAB.BCB.CBA$) have the same set of recurrent factors as the fixed point of $0\mapsto 012$, $1\mapsto 02$, $2\mapsto 1$. The formula $ABAC.BACA.ABCA$ is avoided by polynomially many binary words and there exists arbitrarily many infinite binary words with different sets of recurrent factors that avoid it. If every variable of a ternary formula appears at least twice in the same fragment, then the formula is $3$-avoidable. The pattern $ABACADABCA$ is unavoidable for the class of $C_4$-minor-free graphs with maximum degree~$3$. This disproves a conjecture of Grytczuk. The formula $ABCA.ACBA$, or equivalently the palindromic pattern $ABCADACBA$, has avoidability index $4$.
Unique Rectification in $d$-Complete Posets: Towards the $K$-Theory of Kac-Moody Flag Varieties
Authors: Ilango Rahul , Pechenik Oliver , Zlatin Michael
Published: January 2020

The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of 'unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's '$d$-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of $d$-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain $K$-theoretic Schubert structure constants in the Kac-Moody setting.
$\overline{\cal R}(3,4)=17$
Authors: Prałat Paweł
Published: January 2020

In this paper, we consider the on-line Ramsey numbers $\overline{\cal R} (k,l)$ for cliques. Using a high performance computing networks, we 'calculated' that $\overline{\cal R}(3,4)=17$. We also present an upper bound of $\overline{\cal R}(k,l)$, study its asymptotic behaviour, and state some open problems.
Classifying Rotationally-Closed Languages Having Greedy Universal Cycles
Authors: DiMuro Joseph
Published: January 2020

Let $\textbf{T}(n,k)$ be the set of strings of length $n$ over the alphabet $\Sigma=\{1,2,\ldots,k\}$. A universal cycle for $\textbf{T}(n,k)$ can be constructed using a greedy algorithm: start with the string $k^n$, and continually append the least symbol possible without repeating a substring of length $n$. This construction also creates universal cycles for some subsets $\textbf{S}\subseteq\textbf{T}(n,k)$; we will classify all such subsets that are closed under rotations.
Clumsy Packings of Graphs
Authors: Axenovich Maria , Kaufmann Anika , Yuster Raphael
Published: January 2020

Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains P. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An $H$-packing $P$ is called clumsy if it is maximal of minimum size. Let $\mathrm{cl}(G,H)$ be the size of a clumsy $H$-packing of $G$. We provide nontrivial bounds for $\mathrm{cl}(G,H)$, and in many cases asymptotically determine $\mathrm{cl}(G,H)$ for some generic classes of graphs G such as $K_n$ (the complete graph), $Q_n$ (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine $\mathrm{cl}(K_n,H)$ for every fixed non-empty graph $H$. In particular, we prove that 

$$\mathrm{cl}(K_n, H) = \frac{\binom{n}{2}- \mathrm{ex}(n,H)}{|E(H)|}+o(\mathrm{ex}(n,H)),$$where $ex(n,H)$ is the extremal number of $H$.

A related natural parameter is $\mathrm{cov}(G,H)$, that is the smallest number of copies of $H$ in $G$ (not necessarily edge-disjoint) whose removal from $G$ results in an $H$-free graph. While clearly $\mathrm{cov}(G,H) \leqslant\mathrm{cl}(G,H)$, all of our lower bounds for $\mathrm{cl}(G,H)$ apply to $\mathrm{cov}(G,H)$ as well.
A Note on Counting Flows in Signed Graphs
Authors: DeVos Matt , Rollová Edita , Šámal Robert
Published: January 2020

Tutte initiated  the study of nowhere-zero flows and proved the following fundamental theorem: For every graph $G$ there is a polynomial $f$ so that for every abelian group $\Gamma$ of order $n$, the number of nowhere-zero $\Gamma$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected orientations), the situation is more subtle. For a finite group $\Gamma$, let $\epsilon_2(\Gamma)$ be the largest integer $d$ so that $\Gamma$ has a subgroup isomorphic to $\mathbb{Z}_2^d$. We prove that for every signed graph $G$ and $d \ge 0$ there is a polynomial $f_d$ so that $f_d(n)$ is the number of nowhere-zero $\Gamma$-flows in $G$ for every abelian group $\Gamma$ with $\epsilon_2(\Gamma) = d$ and $|\Gamma| = 2^d n$. Beck and Zaslavsky [JCTB 2006] had previously established the special case of this result when $d=0$ (i.e., when $\Gamma$ has odd order).
A Geometric Interpretation of the Intertwining Number
Authors: Can Mahir Bilen , Cherniavsky Yonah , Rubey Martin
Published: January 2020

We exhibit a connection between two statistics on set partitions, the intertwining number and the depth-index. In particular, results link the intertwining number to the algebraic geometry of Borel orbits. Furthermore, by studying the generating polynomials of our statistics, we determine the $q=-1$ specialization of a $q$-analogue of the Bell numbers. Finally, by using Renner's $H$-polynomial of an algebraic monoid, we introduce and study a $t$-analog of $q$-Stirling numbers.
Greene–Kleitman Invariants for Sulzgruber Insertion
Authors: Garver Alexander , Patrias Rebecca
Published: January 2020

R. Sulzgruber's rim hook insertion and the Hillman–Grassl correspondence are two distinct bijections between the reverse plane partitions of a fixed partition shape and multisets of rim-hooks of the same partition shape. It is known that Hillman–Grassl may be equivalently defined using the Robinson–Schensted–Knuth correspondence, and we show the analogous result for Sulzgruber's insertion. We refer to our description of Sulzgruber's insertion as diagonal RSK. As a consequence of this equivalence, we show that Sulzgruber's map from multisets of rim hooks to reverse plane partitions can be expressed in terms of Greene–Kleitman invariants.
The Ascent-Plateau Statistics on Stirling Permutations
Authors: Ma Shi-Mei , Ma Jun , Yeh Yeong-Nan
Published: January 2020

A permutation $\sigma$ of the multiset $\{1,1,2,2,\ldots,n,n\}$ is called a Stirling permutation of order $n$ if $\sigma_s>\sigma_i$ as long as $\sigma_i=\sigma_j$ and $i<s <j$. In this paper, we present a unified refinement of the ascent polynomials and the ascent-plateau polynomials of Stirling permutations. In particular, by using Foata and Strehl's group action, we prove that the pairs of statistics (left ascent-plateau, ascent) and (left ascent-plateau, plateau) are equidistributed over Stirling permutations of given order, and we show the $\gamma$-positivity of the enumerative polynomial of left ascent-plateaus, double ascents and descent-plateaus. A connection between the $\gamma$-coefficients of this enumerative polynomial and Eulerian numbers is also established.
On the Twelve-Point Theorem for $\ell$-Reflexive Polygons
Authors: Dais Dimitrios I.
Published: January 2020

It is known that, adding the number of lattice points lying on the boundary of a reflexive polygon and the number of lattice points lying on the boundary of its polar, always yields 12. Generalising appropriately the notion of reflexivity, one shows that this remains true for $\ell$-reflexive polygons. In particular, there exist (for this reason) infinitely many (lattice inequivalent) lattice polygons with the same property. The first proof of this fact is due to Kasprzyk and Nill. The present paper contains a second proof (which uses tools only from toric geometry) as well as the description of complementary properties of these polygons and of the invariants of the corresponding toric log del Pezzo surfaces.
On Almost-Equidistant Sets - II
Authors: Polyanskii Alexandr
Published: January 2020

A set in $\mathbb R^d$ is called almost-equidistant if for any three distinct points in the set, some two are at unit distance apart. First, we give a short proof of the result of Bezdek and Lángi claiming that an almost-equidistant set lying on a $(d-1)$-dimensional sphere of radius $r$, where $r<1/\sqrt{2}$, has at most $2d+2$ points. Second, we prove that an almost-equidistant set $V$ in $\mathbb R^d$ has $O(d)$ points in two cases: if the diameter of $V$ is at most $1$ or if $V$ is a subset of a $d$-dimensional ball of radius at most $1/\sqrt{2}+cd^{-2/3}$, where $c<1/2$. Also, we present a new proof of the result of Kupavskii, Mustafa and Swanepoel that an almost-equidistant set in $\mathbb R^d$ has $O(d^{4/3})$ elements.
Irregularity Strength of Regular Graphs
Authors: Przybyło Jakub
Published: January 2020

Let $G$ be a simple graph with no isolated edges and at most one isolated vertex. For a positive integer $w$, a $w$-weighting of $G$ is a map $f:E(G)\rightarrow \{1,2,\ldots,w\}$. An irregularity strength of $G$, $s(G)$, is the smallest $w$ such that there is a $w$-weighting of $G$ for which $\sum_{e:u\in e}f(e)\neq\sum_{e:v\in e}f(e)$ for all pairs of different vertices $u,v\in V(G)$. A conjecture by Faudree and Lehel says that there is a constant $c$ such that $s(G)\le{n\over d}+c$ for each $d$-regular graph $G$, $d\ge 2$. We show that $s(G) < 16{n\over d}+6$. Consequently, we improve the results by Frieze, Gould, Karoński and Pfender (in some cases by a $\log n$ factor) in this area, as well as the recent result by Cuckler and Lazebnik.
On Certain Combinatorial Expansions of the Legendre-Stirling Numbers
Authors: Ma Shi-Mei , Ma Jun , Yeh Yeong-Nan
Published: January 2020

The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. As a continuation of the work of Andrews and Littlejohn (Proc. Amer. Math. Soc., 137 (2009), 2581-2590), we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain expansions of the Legendre-Stirling numbers of both kinds in terms of binomial coefficients.
Off-Diagonal Ordered Ramsey Numbers of Matchings
Authors: Rohatgi Dhruv
Published: January 2020

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete ordered graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the off-diagonal ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that there is some fixed $\epsilon > 0$ such that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for ordered matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random ordered matchings with interval chromatic number $2$.
On Grounded L-Graphs and their Relatives
Authors: Jelínek Vít , Töpfer Martin
Published: January 2020

We consider the graph classes Grounded-L and Grounded-{𝖫,⅃} corresponding to graphs that admit an intersection representation by 𝖫-shaped curves (or 𝖫-shaped and ⅃-shaped curves, respectively), where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns.
We also compare these classes to related intersection classes, such as the grounded segment graphs, the monotone 𝖫-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinct and satisfy only trivial or previously known inclusions.
Nonexistence of Almost Moore Digraphs of Diameter Three
Authors: Conde J. , Gimbert J. , Gonzàlez J. , Miret J. M. , Moreno R.
Published: January 2020

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound $M(d,k)=1+d+\cdots +d^k$, where $d>1$ and $k>1$ denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when $d=2,3$ or $k=2$. In this paper, we prove that almost Moore digraphs of diameter $k=3$ do not exist for any degree $d$. The enumeration of almost Moore digraphs of degree $d$ and diameter $k=3$ turns out to be equivalent to the search of binary matrices $A$ fulfilling that $AJ=dJ$ and $I+A+A^2+A^3=J+P$, where $J$ denotes the all-one matrix and $P$ is a permutation matrix. We use spectral techniques in order to show that such equation has no $(0,1)$-matrix solutions. More precisely, we obtain the factorization in ${\Bbb Q}[x]$ of the characteristic polynomial of $A$, in terms of the cycle structure of $P$, we compute the trace of $A$ and we derive a contradiction on some algebraic multiplicities of the eigenvalues of $A$. In order to get the factorization of $\det(xI-A)$ we determine when the polynomials $F_n(x)=\Phi_n(1+x+x^2+x^3)$ are irreducible in ${\Bbb Q}[x]$, where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial, since in such case they become 'big pieces' of $\det(xI-A)$. By using concepts and techniques from algebraic number theory, we prove that $F_n(x)$ is always irreducible in ${\Bbb Q}[x]$, unless $n=1,10$. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.
Asymptotics of Coefficients of Multivariate Generating Functions: Improvements for Smooth Points
Authors: Raichev Alexander , Wilson Mark C.
Published: January 2020

Let $\sum_{\beta\in{\Bbb N}^d} F_\beta x^\beta$ be a multivariate power series. For example $\sum F_\beta x^\beta$ could be a generating function for a combinatorial class. Assume that in a neighbourhood of the origin this series represents a nonentire function $F=G/H^p$ where $G$ and $H$ are holomorphic and $p$ is a positive integer. Given a direction $\alpha\in{\Bbb N}_+^d$ for which the asymptotics are controlled by a smooth point of the singular variety $H = 0$, we compute the asymptotics of $F_{n \alpha}$ as $n\to\infty$. We do this via multivariate singularity analysis and give an explicit uniform formula for the full asymptotic expansion. This improves on earlier work of R. Pemantle and the second author and allows for more accurate numerical approximation, as demonstrated by our our examples (on lattice paths, quantum random walks, and nonoverlapping patterns).
Stability Results for Vertex Turán Problems in Kneser Graphs
Authors: Gerbner Dániel , Methuku Abhishek , Nagy Dániel T. , Patkos Balazs , Vizer Máté
Published: January 2020

The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free, was recently determined by Alishahi and Taherkhani, whenever $n$ is large enough compared to $k$ and $F$. In this paper, we determine the second largest size of a vertex set $W \subseteq V$ such that $K(n,k)[W]$ is $F$-free, in the case when $F$ is an even cycle or a complete multi-partite graph. In the latter case, we actually give a more general theorem depending on the chromatic number of $F$.
Permutations in Two Dimensions that Maximally Separate Neighbors
Authors: Albow Mohammed , Edgington Jeffrey , Lopez Mario , Vojtěchovský Petr
Published: January 2020

We characterize all permutations on even-by-even grids that maximally separate neighboring vertices. More precisely, let $n_1$, $n_2$ be positive even integers, let $I(n_1,n_2)=\{1,\dots,n_1\}\times\{1,\dots,n_2\}$ be the $n_1\times n_2$ grid, let $\mathrm{d}$ be the $L_1$ metric on $I(n_1,n_2)$, and let $N=\{\{x,y\}\in I(n_1,n_2)\times I(n_1,n_2): \mathrm{d}(x,y)=1\}$ be the set of neighbors in $I(n_1,n_2)$. We characterize all permutations $\pi$ of $I(n_1,n_2)$ that maximize $\sum_{\{x,y\}\in N} \mathrm{d}(\pi(x),\pi(y))$.
Super RSK Correspondence with Symmetry
Authors: Muth Robert
Published: January 2020

Super RSK correspondence is a bijective correspondence between superbiwords and pairs of semistandard supertableaux. Such a bijection was given by Bonetti, Senato and Venezia, via an insertion algorithm closely related to Schensted insertion. Notably, the symmetry property satisfied by the classical RSK bijection holds only in special cases under this bijection. We present a new super RSK bijection, based on the mixed insertion process defined by Haiman, where the symmetry property holds in complete generality.
On the Schur Function Expansion of a Symmetric Quasi-symmetric Function
Authors: Gessel Ira M.
Published: January 2020

Egge, Loehr, and Warrington proved a formula for the Schur function expansion of a symmetric function in terms of its expansion in fundamental quasi-symmetric functions. Their formula involves the coefficients of a modified inverse Kostka matrix. Recently Garsia and Remmel gave a simpler reformulation of Egge, Loehr, and Warrington's result, with a new proof. We give here a simple proof of Garsia and Remmel's version, using a sign-reversing involution.
Spectral Lower Bounds for the Orthogonal and Projective Ranks of a Graph
Authors: Wocjan Pawel , Elphick Clive
Published: January 2020

The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $\xi$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^\xi$ for $v\in V$ satisfying the orthogonality condition $x_v^\dagger x_w=0$ for all $vw\in E$.   We prove that many spectral lower bounds for the chromatic number, $\chi$, are also lower bounds for $\xi$. This result complements a previous result by the authors, in which they showed that spectral lower bounds for $\chi$ are also lower bounds for the quantum chromatic number $\chi_q$.  It is known that the quantum chromatic number and the orthogonal rank are incomparable.  

We conclude by proving an inertial  lower bound for the projective rank $\xi_f$, and conjecture that a stronger inertial lower bound for $\xi$ is also a lower bound for $\xi_f$.
The Smallest Strictly Neumaier Graph and its Generalisations
Authors: Evans Rhys J. , Goryainov Sergey , Panasenko Dmitry
Published: January 2020

A regular clique in a regular graph is a clique such that every vertex outside of the clique is adjacent to the same positive number of vertices inside the clique. We continue the study of regular cliques in edge-regular graphs initiated by A. Neumaier in the 1980s and attracting current interest. We thus define a Neumaier graph to be an non-complete edge-regular graph containing a regular clique, and a strictly Neumaier graph to be a non-strongly regular Neumaier graph. We first prove some general results on Neumaier graphs and their feasible parameter tuples. We then apply these results to determine the smallest strictly Neumaier graph, which has $16$ vertices. Next we find the parameter tuples for all strictly Neumaier graphs having at most $24$ vertices. Finally, we give two sequences of graphs, each with $i^{th}$ element a strictly Neumaier graph containing a $2^{i}$-regular clique (where $i$ is a positive integer) and having parameters of an affine polar graph as an edge-regular graph. This answers questions recently posed by G. Greaves and J. Koolen.
Non-Monochromatic Triangles in a 2-Edge-Coloured Graph
Authors: DeVos Matt , McDonald Jessica , Montejano Amanda
Published: January 2020

Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$.  We prove that whenever $|E| + \mathrm{min}\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$. If instead equality holds, and $G$ has no such subgraph, then we show that $G$ is in one of a few simple classes.
Cyclic Sieving, Necklaces, and Branching Rules Related to Thrall's Problem
Authors: Ahlbach Connor , Swanson Joshua P.
Published: January 2020

We show that the cyclic sieving phenomenon of Reiner-Stanton-White together with necklace generating functions arising from work of Klyachko offer a remarkably unified, direct, and largely bijective approach to a series of results due to Kraśkiewicz-Weyman, Stembridge, and Schocker related to the so-called higher Lie modules and branching rules for inclusions $ C_a \wr S_b \hookrightarrow S_{ab} $. Extending the approach gives monomial expansions for certain graded Frobenius series arising from a generalization of Thrall's problem.
Tree-Like Distance Colouring for Planar Graphs of Sufficient Girth
Authors: Kang Ross J. , Van Loon Willem
Published: January 2020

Given a multigraph $G$ and a positive integer $t$, the distance-$t$ chromatic index of $G$ is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than $t$ edges must receive different colours. Let $\pi'_t(d)$ and $\tau'_t(d)$ be the largest values of this parameter over the class of planar multigraphs and of (simple) trees, respectively, of maximum degree $d$. We have that $\pi'_t(d)$ is at most and at least a non-trivial constant multiple larger than $\tau'_t(d)$. (We conjecture $\limsup_{d\to\infty}\pi'_2(d)/\tau'_2(d) =9/4$ in particular.) We prove for odd $t$ the existence of a quantity $g$ depending only on $t$ such that the distance-$t$ chromatic index of any planar multigraph of maximum degree $d$ and girth at least $g$ is at most $\tau'_t(d)$ if $d$ is sufficiently large. Such a quantity does not exist for even $t$. We also show a related, similar phenomenon for distance vertex-colouring.
Permutation Graphs and the Abelian Sandpile Model, Tiered Trees and Non-Ambiguous Binary Trees
Authors: Dukes Mark , Selig Thomas , Smith Jason P. , Steingrímsson Einar
Published: January 2020

A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. We show that the set of minimal recurrent configurations is in bijection with the set of complete non-ambiguous binary trees introduced by Aval et al. [2], and introduce a multi-rooted generalization of these that we show to correspond to all recurrent configurations. In the case of permutations with a single descent, we recover some results from the case of Ferrers graphs presented in [11], while we also recover results of Perkinson et al. [16] in the case of threshold graphs.
Split Graphs: Combinatorial Species and Asymptotics
Authors: Troyka Justin M.
Published: January 2020

A split graph is a graph whose vertices can be partitioned into a clique and a stable set. We investigate the combinatorial species of split graphs, providing species-theoretic generalizations of enumerative results due to Bína and Přibil (2015), Cheng, Collins, and Trenk (2016), and Collins and Trenk (2018). In both the labeled and unlabeled cases, we give asymptotic results on the number of split graphs, of unbalanced split graphs, and of bicolored graphs, including proving the conjecture of Cheng, Collins, and Trenk (2016) that almost all split graphs are balanced.
On the Spectral Characterization of Mixed Extensions of $P_{3}$
Authors: Haemers Willem H. , Sorgun Sezer , Topcu Hatice
Published: January 2020

A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by replacing each vertex of $G$ by a clique or a coclique, whilst two vertices in $H$ corresponding to distinct vertices $x$ and $y$ of $G$ are adjacent whenever $x$ and $y$ are adjacent in $G$. If $G$ is the path $P_3$, then $H$ has at most three adjacency eigenvalues unequal to $0$ and $-1$. Recently, the first author classified the graphs with the mentioned eigenvalue property. Using this classification we investigate mixed extension of $P_3$ on being determined by the adjacency spectrum. We present several cospectral families, and with the help of a computer we find all graphs on at most 25 vertices that are cospectral with a mixed extension of $P_3$.
The Problem of Pawns
Authors: Brown Tricia Muldoon
Published: January 2020

Using a bijective proof, we show the number of ways to arrange a maximum number of nonattacking pawns on a $2m\times 2m$ chessboard is ${2m\choose m}^2$, and more generally, the number of ways to arrange a maximum number of nonattacking pawns on a $2n \times 2m$ chessboard is ${m+n\choose n}^2$.
Tutte Polynomial, Subgraphs, Orientations and Sandpile Model: New Connections via Embeddings
Authors: Bernardi Olivier
Published: January 2020

We define a bijection between spanning subgraphs and orientations of graphs and explore its enumerative consequences regarding the Tutte polynomial. We obtain unifying bijective proofs for all the evaluations $T_G(i,j),0\leq i,j \leq 2$ of the Tutte polynomial in terms of subgraphs, orientations, outdegree sequences and sandpile configurations. For instance, for any graph $G$, we obtain a bijection between connected subgraphs (counted by $T_G(1,2)$) and root-connected orientations, a bijection between forests (counted by $T_G(2,1)$) and outdegree sequences and bijections between spanning trees (counted by $T_G(1,1)$), root-connected outdegree sequences and recurrent sandpile configurations. All our proofs are based on a single bijection $\Phi$ between the spanning subgraphs and the orientations that we specialize in various ways. The bijection $\Phi$ is closely related to a recent characterization of the Tutte polynomial relying on combinatorial embeddings of graphs, that is, on a choice of cyclic order of the edges around each vertex.
Cutting Resilient Networks – Complete Binary Trees
Authors: Cai Xing Shi , Holmgren Cecilia
Published: January 2020

In our previous work, we introduced the random $k$-cut number for rooted graphs. In this paper, we show that the distribution of the $k$-cut number in complete binary trees of size n, after rescaling, is asymptotically a periodic function of $\lg n − \lg \lg n$. Thus there are different limit distributions for different subsequences, where these limits are similar to weakly $1$-stable distributions. This generalizes the result for the case $k = 1$, i.e., the traditional cutting model, by Janson (2004).
Collapsibility of Simplicial Complexes of Hypergraphs
Authors: Lew Alan
Published: January 2020

Let $\mathcal{H}$ be an $r$-uniform hypergraph. We show that the simplicial complex whose simplices are the hypergraphs $\mathcal{F}\subset\mathcal{H}$ with covering number at most $p$ is $\left(\binom{r+p}{r}-1\right)$-collapsible. Similarly, the simplicial complex whose simplices are the pairwise intersecting hypergraphs $\mathcal{F}\subset\mathcal{H}$ is $\frac{1}{2}\binom{2r}{r}$-collapsible.
A Note on the Connectivity of 2-Polymatroid Minors
Authors: Gershkoff Zachary , Oxley James
Published: January 2020

Brylawski and Seymour independently proved that if $M$ is a connected matroid with a connected minor $N$, and $e \in E(M) - E(N)$, then $M \backslash e$ or $M / e$ is connected having $N$ as a minor. This paper proves an analogous but somewhat weaker result for $2$-polymatroids. Specifically, if $M$ is a connected $2$-polymatroid with a proper connected minor $N$, then there is an element $e$ of $E(M) - E(N)$ such that $M \backslash e$ or $M / e$ is connected having $N$ as a minor. We also consider what can be said  about the uniqueness of the way in which the elements of $E(M) - E(N)$ can be removed so that connectedness is always maintained.
Partitioning the Power Set of $[n]$ into $C_k$-free parts
Authors: Blaisdell Eben , Gyárfás András , Krueger Robert A. , Wdowinski Ronen
Published: January 2020

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle — three different subsets $A,B,C\subseteq [n]$ such that $A\cap B,A\cap C,B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts.  We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp when $G$ is a cycle, path, or star. Additional bounds are given when $G$ is a $4$-cycle and when $G$ is a claw.
On Subsequence Sums of a Zero-sum Free Sequence II
Authors: Gao Weidong , Li Yuanlin , Peng Jiangtao , Sun Fang
Published: January 2020

Let $G$ be an additive finite abelian group with exponent $\exp (G) = n$. For a sequence $S$ over $G$, let f$(S)$ denote the number of non-zero group elements which can be expressed as a sum of a nontrivial subsequence of $S$. We show that for every zero-sum free sequence $S$ over $G$ of length $|S| = n+1$ we have f$(S) \ge 3n-1$.
On the Chromatic Number of Simple Triangle-Free Triple Systems
Authors: Frieze Alan , Mubayi Dhruv
Published: January 2020

A hypergraph is simple if every two edges share at most one vertex. It is triangle-free if in addition every three pairwise intersecting edges have a vertex in common. We prove that there is an absolute constant $c$ such that the chromatic number of a simple triangle-free triple system with maximum degree $\Delta$ is at most $c\sqrt{\Delta/\log \Delta}$. This extends a result of Johansson about graphs, and is sharp apart from the constant $c$.
Coordinate Deletion of Zeroes
Authors: Räty Eero
Published: January 2020

For a family $A\subseteq\left\{ 0,\dots,k\right\} ^{n}$, define the $\delta$-shadow of $A$ to be the set obtained from $A$ by removing from any of its vectors one coordinate that equals zero. Given the size of $A$, how should we choose $A$ to minimise its $\delta$-shadow? Our aim in this paper is to show that, for any $r$, the family of all sequences with at most $r$ zeros has minimal $\delta$-shadow. We actually give the exact best $A$ for every size.
Fractional Factors, Component Factors and Isolated Vertex Conditions in Graphs
Authors: Yu Roger , Kano Mikio , Lu Hongliang
Published: January 2020

For a graph $G = (V, E)$, a fractional $[a, b]$-factor is a real valued function $h:E(G)\to [0,1]$ that satisfies $a \le ~ \sum_{e\in E_G(v)} h(e) ~ \le b$ for all $ v\in V(G)$, where $a$ and $b$ are real numbers and $E_G(v)$ denotes the set of edges incident with $v$. In this paper, we prove that the condition $\mathit{iso}(G-S) \le (k+\frac{1}{2})|S|$ is equivalent to the existence of fractional $[1,k+ \frac{1}{2}]$-factors, where ${\mathit{iso}}(G-S)$ denotes the number of isolated vertices in $G-S$. Using fractional factors as a tool, we construct component factors under the given isolated conditions. Namely, (i) a graph $G$ has a $\{P_2,C_3,P_5, \mathcal{T}(3)\}$-factor if and only if $\mathit{iso}(G-S) \le \frac{3}{2}|S|$ for all $S\subset V(G)$; (ii) a graph $G$ has a $\{K_{1,1}, K_{1,2}, \ldots,$ $K_{1,k}, \mathcal{T}(2k+1)\}$-factor ($k\ge 2$) if and only if $\mathit{iso}(G-S) \le (k+\frac{1}{2})|S|$ for all $S\subset V(G)$, where $\mathcal{T}(3)$ and $\mathcal{T}(2k+1)$ are two special families of trees.
$(\ell,0)$-Carter Partitions, their Crystal-Theoretic Behavior and Generating Function
Authors: Berg Chris , Vazirani Monica
Published: January 2020

In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions". The representation-theoretic significance of these partitions is that they indicate the irreducibility of the corresponding specialized Specht module over the Hecke algebra of the symmetric group. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas, which is in terms of hook lengths. We use our result to find a generating series which counts such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\Lambda_0)$ of the basic representation of $\widehat{{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $(\ell,0)$-Carter partitions in the graph $B(\Lambda_0)$.