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A standard barely set-valued tableau of shape $\lambda$ is a filling of the Young diagram $\lambda$ with integers $1,2,\dots,|\lambda|+1$ such that the integers are increasing in each row and column, and every cell contains one integer except one cell that contains two integers. Counting standard barely set-valued tableaux is closely related to the coincidental down-degree expectations (CDE) of lower intervals in Young's lattice. Using $q$-integral techniques we give a formula for the number of standard barely set-valued tableaux of arbitrary shifted shape. We show how it can be used to recover two formulas, originally conjectured by Reiner, Tenner and Yong, and proved by Hopkins, for numbers of standard barely set valued tableaux of particular shifted-balanced shapes. We also prove a conjecture of Reiner, Tenner and Yong on the CDE property of the shifted shape $(n,n-2,n-4,\dots,n-2k+2)$. Finally, in the Appendix we raise a conjecture on an $\mathsf a;q$-analogue of the down-degree expectation with respect to the uniform distribution for a specific class of lower order ideals of Young's lattice., Comment: 24 pages, 22 figures This is a revised version of the previous paper. Remark 5.5 is added; Conjecture A.2 in the Appendix is strengthened; with further minor changes

Let $n\ge 1$, $r\ge 2$, and $s\ge 0$ be integers and ${\cal P}=\{P_1,\dots, P_l\}$ be a partition of $[n]=\{1,\dots, n\}$ with $|P_i|\le r$ for $i=1,\dots, l$. Also, let $\cal F$ be a family of non-empty subsets of $[n]$. The $r$-uniform Kneser-type hypergraph $\mbox{KG}^r({\cal F}, {\cal P},s)$ is the hypergraph with the vertex set of all $\cal P$-admissible elements $F\in {\cal F}$, that is $|F\cap P_i|\le 1$ for $i=1,\dots, l$ and the edge set of all $r$-subsets $\{F_1,\dots, F_r\}$ of the vertex set that $|F_i\cap F_j|\le s$ for all $1\le i, Comment: An error in the proof of the main theorem is fixed by weakening the statement of the theorem. Typos are fixed throughout the text

The Burling sequence is a sequence of triangle-free graphs of increasing chromatic number. Each of them is isomorphic to the intersection graph of a set of axis-parallel boxes in $R^3$. These graphs were also proved to have other geometrical representations: intersection graphs of line segments in the plane, and intersection graphs of frames, where a frame is the boundary of an axis-aligned rectangle in the plane. We call Burling graph every graph that is an induced subgraph of some graph in the Burling sequence. We give five new equivalent ways to define Burling graphs. Three of them are geometrical, one is of a more graph-theoretical flavour and one is more axiomatic., 35 pages, 20 figures

The clustering of a graph coloring is the maximum size of monochromatic components. This paper studies colorings with bounded clustering in graph classes with bounded layered treewidth, which include planar graphs, graphs of bounded Euler genus, graphs embeddable on a fixed surface with a bounded number of crossings per edge, map graphs, amongst other examples. Our main theorem says that every graph with layered treewidth at most $k$ and with maximum degree at most $\Delta$ is $3$-colorable with clustering $O(k^{19}\Delta^{37})$. This is the first known polynomial bound on the clustering. This greatly improves upon a corresponding result of Esperet and Joret for graphs of bounded genus., Comment: This paper is extracted from arXiv:1905.08969. The corresponding part in arXiv:1905.08969 will be deleted

Motivated by the counting results for color-critical subgraphs by Mubayi [Adv. Math., 2010], we study the phenomenon behind Mubayi's theorem from a spectral perspective and start up this problem with the fundamental case of triangles. We prove tight bounds on the number of copies of triangles in a graph with a prescribed number of vertices and edges and spectral radius. Let $n$ and $m$ be the order and size of a graph. Our results extend those of Nosal, who proved there is one triangle if the spectral radius is more than $\sqrt{m}$, and of Rademacher, who proved there are at least $\lfloor\frac{n}{2}\rfloor$ triangles if the number of edges is more than that of 2-partite Tur\'an graph. These results, together with two spectral inequalities due to Bollob\'as and Nikiforov, can be seen as a solution to the case of triangles of a problem of finding spectral versions of Mubayi's theorem. In addition, we give a short proof of the following inequality due to Bollob\'as and Nikiforov [J. Combin. Theory Ser. B, 2007]: $t(G)\geq \frac{\lambda(G)(\lambda^2(G)-m)}{3}$ and characterize the extremal graphs. Some problems are proposed in the end., Comment: 15 pages; 1 figure; 1 table

We show that if $G$ is a $d$--regular graph on $n$ vertices, then the number of spanning forests $F(G)$ satisfies $F(G)\leq d^n$. The previous best bound due to Kahale and Schulman gave $(d+1/2+O(1/d))^n$. We also have the more precise conjecture that $$F(G)^{1/n}\leq \frac{(d-1)^{d-1}}{(d^2-2d-1)^{d/2-1}}.$$ If this conjecture is true, then the expression on the right hand side is the best possible., arXiv admin note: text overlap with arXiv:2105.06798

We provide lower and upper bounds on the minimum size of a maximum stable set over graphs of flag spheres, as a function of the dimension of the sphere and the number of vertices. Further, we use stable sets to obtain an improved Lower Bound Theorem for the face numbers of flag spheres., Comment: 10 pages, 2 figures, to appear at FPSAC2022. Major revision, in particular, Thm.1.3 now claims a weaker lower bound, as in v1 the proof had a gap: we wrongly claimed that a certain sum of ideals is direct. These ideals are irrelevant in the current version

The neighborhood complex of a graph was introduced by Lov\'asz to provide topological lower bounds on chromatic number, and more general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such `Hom complexes' are also related to reconfiguration problems and a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs, which have applications in the study of graded posets and resolutions of monomial ideals. Our construction can be seen as a special case of the poset structure on the set of multihomomorphisms in more general categories, as introduced by Kozlov, Matsushita, and others. We relate the topological properties of Hom complexes to certain digraph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a directed graph and prove that its homotopy type is recovered as a certain Hom complex. We establish a number of results regarding the topology of these complexes, including the dependence on directed bipartite subgraphs, a directed version of the Mycielksi construction, as well as vanishing theorems for higher homology. Inspired by notions of reconfigurations of directed graph colorings we study the connectivity of Hom complexes into tournaments $T_n$. We obtain a complete answer for the case of transitive $T_n$, where we also describe a connection to mixed subdivisions of dilated simplices. Finally we use paths in the internal hom objects of directed graphs to define various notions of homotopy, and discuss connections to the topology of Hom complexes., Comment: 34 pages, 10 figures

Let $\mathcal G$ be a family of subsets of an $n$-element set. The family $\mathcal G$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in $\mathcal G$ is of size at least $t$. For a real number $p\in(0,1)$ we define the measure of the family by the sum of $p^{|G|}(1-p)^{n-|G|}$ over all $G\in\mathcal G$. We prove that if $t\geq 15$ and $p\leq 2/(\sqrt{4t+9}-1)$ then $p^t$ is the maximum measure of $3$-wise $t$-intersecting families, and the bound for $p$ is sharp. We also present the corresponding stability result for shifted families.

A subtree of a tree is any induced subgraph that is again a tree (i.e., connected). The mean subtree order of a tree is the average number of vertices of its subtrees. This invariant was first analyzed in the 1980s by Jamison. An intriguing open question raised by Jamison asks whether the maximum of the mean subtree order, given the order of the tree, is always attained by some caterpillar. While we do not completely resolve this conjecture, we find some evidence in its favor by proving different features of trees that attain the maximum. For example, we show that the diameter of a tree of order $n$ with maximum mean subtree order must be very close to $n$. Moreover, we show that the maximum mean subtree order is equal to $n - 2\log_2 n + O(1)$. For the local mean subtree order, which is the average order of all subtrees containing a fixed vertex, we can be even more precise: we show that its maximum is always attained by a broom and that it is equal to $n - \log_2 n + O(1)$., Comment: 21 pages, 3 figures

We show that for every graph $G$ that contains two edge-disjoint spanning trees, we can choose two edge-disjoint spanning trees $T_1,T_2$ of $G$ such that $|d_{T_1}(v)-d_{T_2}(v)|\leq 5$ for all $v \in V(G)$. We also prove the more general statement that for every positive integer $k$, there is a constant $c_k \in O(\log k)$ such that for every graph $G$ that contains $k$ edge-disjoint spanning trees, we can choose $k$ edge-disjoint spanning trees $T_1,\ldots,T_k$ of $G$ satisfying $|d_{T_i}(v)-d_{T_j}(v)|\leq c_k$ for all $v \in V(G)$ and $i,j \in \{1,\ldots,k\}$. This resolves a conjecture of Kriesell.

The aim of this work is to use linear programming and polyhedral geometry to prove a duality formula for the ic-resurgence of edge ideals. We show that the ic-resurgence of the edge ideal $I$ of a clutter $\mathcal{C}$ and the ic-resurgence of the edge ideal $I^\vee$ of the blocker $\mathcal{C}^\vee$ of $\mathcal{C}$ coincide. If $\mathcal{C}$ is the clutter of bases of certain uniform matroids, we recover a formula for the resurgence of $I$, and if $\mathcal{C}$ is a connected non-bipartite graph with a perfect matching, we show a formula for the Waldschmidt constant of $I^\vee$.

In this paper we introduce {\em weak ascent sequences}, a class of number sequences that properly contains ascent sequences. We show how these sequences uniquely encode each of the following objects: permutations avoiding a particular length-4 bivincular pattern; upper-triangular binary matrices that satisfy a column-adjacency rule; factorial posets that are weakly (3+1)-free. We also show how weak ascent sequences are related to a class of pattern avoiding inversion sequences that has been a topic of recent research by Auli and Elizalde. Finally, we consider the problem of enumerating these new sequences and give a closed form expression for the number of weak ascent sequences having a prescribed length and number of weak ascents.

A fundamental and challenging problem in spectral graph theory is to characterize which graphs are uniquely determined by their spectra. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author proved that an $n$-vertex graph $G$ is uniquely determined by its generalized spectrum (DGS) whenever $2^{-\lfloor\frac{n}{2}\rfloor}\det W$ is odd and square-free. Here, $W$ is the walk matrix of $G$, namely, $W=[e,Ae,\ldots,A^{n-1}e]$ with $e$ all-one vector and $A$ the adjacency matrix of $G$. In this paper, we focus on a larger family of graphs with $d_n$ square-free, where $d_n$ refers to the last invariant factor of $W$. We introduce a new kind of polynomials for a graph $G$ associated with a prime $p$. Such a polynomial is invariant under generalized cospectrality. Using the newly defined polynomials, we obtain a sufficient condition for a graph in the larger family to be DGS. The main result of this paper improves upon the aforementioned result of Wang while the proof for the main result gives a new way to attack the problem of generalized spectral characterization of graphs., 14 pages

We prove that the parameter $x$ of a tight set $\mathcal{T}$ of a hyperbolic quadric $\mathsf{Q}^+(2n+1,q)$ of an odd rank $n+1$ satisfies ${x\choose 2}+w(w-x)\equiv 0\mod q+1$, where $w$ is the number of points of $\mathcal{T}$ in any generator of $\mathsf{Q}^+(2n+1,q)$. As this modular equation should have an integer solution in $w$ if such a $\mathcal{T}$ exists, this condition rules out roughly at least one half of all possible parameters $x$. It generalizes a previous result by the author and K. Metsch shown for tight sets of a hyperbolic quadric $\mathsf{Q}^+(5,q)$ (also known as Cameron-Liebler line classes in $\mathrm{PG}(3,q)$).

We take the first steps in developing a theory of transport of patterns from Fishburn permutations to (modified) ascent sequences. Given a set of pattern avoiding Fishburn permutations, we provide an explicit construction for the basis of the corresponding set of modified ascent sequences. Our approach is in fact more general and can transport patterns between permutations and equivalence classes of so called Cayley permutations. This transport of patterns relies on a simple operation we call the Burge transpose. It operates on certain biwords called Burge words. Moreover, using mesh patterns on Cayley permutations, we present an alternative view of the transport of patterns as a Wilf-equivalence between subsets of Cayley permutations. We also highlight a connection with primitive ascent sequences., 30 pages, 6 figures. To appear in the European Journal of Combinatorics

In this article we develop a general method by which one can explicitly evaluate certain sums of $n$-th powers of products of $d\geq 1$ elementary trigonometric functions evaluated at $\mathbf{m}=(m_1,\ldots,m_d)$-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a $d$-dimensional discrete torus $G_{\mathbf{m}}$ which depends on $\mathbf{m}$. The sums in question are then related to the $n$-th step of a Markov chain on $G_{\mathbf{m}}$. The Markov chain admits the interpretation as a particular random walk, also viewed as a discrete time and discrete space heat diffusion, so then the sum in question is related to special values of the associated heat kernel. Our evaluation follows by deriving a combinatorial expression for the heat kernel, which is obtained by periodizing the heat kernel on the infinite lattice $\mathbb{Z}^{d}$ which covers $G_{\mathbf{m}}$.

The chromatic symmetric function (CSF) of Dyck paths of Stanley and its Shareshian-Wachs $q$-analogue have important connections to Hessenberg varieties, diagonal harmonics and LLT polynomials. In the, so called, abelian case they are also curiously related to placements of non-attacking rooks by results of Stanley-Stembridge (1993) and Guay-Paquet (2013). For the $q$-analogue, these results have been generalized by Abreu-Nigro (2020) and Guay-Paquet (private communication), using $q$-hit numbers. Among our main results is a new proof of Guay-Paquet's elegant identity expressing the $q$-CSFs in a CSF basis with $q$-hit coefficients. We further show its equivalence to the Abreu-Nigro identity expanding the $q$-CSF in the elementary symmetric functions. In the course of our work we establish that the $q$-hit numbers in these expansions differ from the originally assumed Garsia-Remmel $q$-hit numbers by certain powers of $q$. We prove new identities for these $q$-hit numbers, and establish connections between the three different variants., full version, 41 pages, 7 figures. arXiv admin note: text overlap with arXiv:2012.00913; v2 fixed typos, corrected proof of Prop. 6.5, updated figures and references

We give a new proof of the Skeletal Lemma, which is the main technical tool in our paper on Hamilton cycles in line graphs [T. Kaiser and P. Vr\'ana, Hamilton cycles in 5-connected line graphs, European J. Combin. 33 (2012), 924-947]. It generalises results on disjoint spanning trees in graphs to the context of 3-hypergraphs. The lemma is proved in a slightly stronger version that is more suitable for applications. The proof is simplified and formulated in a more accessible way.

The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is often the case that every two-edge-coloring of the complete graph on $r(H)$ vertices contains many monochromatic copies of $H$. The minimum number of such copies over all two-colorings of $K_{r(H)}$ will be referred to as the threshold Ramsey multiplicity of $H$. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant $c$ such that the threshold Ramsey multiplicity of a path or an even cycle on $k$ vertices is at least $(ck)^k$. This bound is tight up to the constant $c$. We prove a similar result for odd cycles in a companion paper., 28 pages

We investigate the class of infinite distance-transitive digraphs $D$ of finite out-valency. We show that if $D$ is a weakly descendant-homogeneous in such a class then either (1) $D$ has property $Z$ and the reachability relation is not universal; or (2) $D$ does not have property $Z$, the reachability relation is universal and $D$ has infinite in-valency. Also, we show that earlier results, proved in the context of highly-arc-transitive digraphs, hold under the weaker condition of distance-transitivity. Finally, we give a description of a subclass of the class distance-transitive weakly descendant-homogeneous digraphs for which the reachability relation is not universal.

We give the facet description of the deformation cones of graph associahedra and nestohedra, generalizing the classical parametrization of the family of deformed permutahedra by the cone of submodular functions. When the underlying building set is made of intervals, this leads in particular to the construction of kinematic nestohedra generalizing the kinematic associahedra that recently appeared in the theory of scattering amplitudes., Comment: 27 pages, 10 figures. Section 2 is adapted from Section 2.4 of arXiv:1906.06861v2 (and will disappear there)

Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety of a classical group. We provide a combinatorial description for these polynomials, as well as their double versions, by introducing analogues of pipe dreams, or RC-graphs, for the Weyl groups of the classical types., 36 pages, 10 figures. v2: appendix added. v3: Sec 4 added. Final version, to appear in European Journal of Combinatorics

We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median Genocchi numbers. By establishing a bijection with objects defined by Dumont, we show that alternation acyclic tournaments in which at least one ascent begins at each vertex, except for the largest one, are counted by the Genocchi numbers of the first kind. Unexpected consequences of our results include a pair of ordinary generating function formulas for the Genocchi numbers of both kinds and a new very simple model for the normalized median Genocchi numbers., Comment: Minor but important corrections. This paper has been accepted by the European Journal of Combinatorics

We apply the homomorphism complex construction to partially ordered sets, introducing a new topological construction based on the set of maximal chains in a graded poset. Our primary objects of study are distributive lattices, with special emphasis on finite products of chains. For the special case of a Boolean algebra, we observe that the corresponding homomorphism complex is isomorphic to the subcomplex of cubical cells in a permutahedron. Thus, this work can be interpreted as a generalization of the study of these complexes. We provide a detailed investigation when our poset is a product of chains, in which case we find an optimal discrete Morse matching and prove that the corresponding complex is torsion-free., Comment: the first version was missing a statement that Kozlov defined the general hom complex construction

For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $\sigma$ such that every set of $t-1$ consecutive elements in $\sigma$ is contained in a $t$-element circuit and $t$-element cocircuit. We show that if $M$ has the cyclic $(t-1,t)$-property and $|E(M)|$ is sufficiently large, then these $t$-element circuits and $t$-element cocircuits are arranged in a prescribed way in $\sigma$, which, for odd $t$, is analogous to how 3-element circuits and cocircuits appear in wheels and whirls, and, for even $t$, is analogous to how 4-element circuits and cocircuits appear in swirls. Furthermore, we show that any appropriate concatenation $\Phi$ of $\sigma$ is a flower. If $t$ is odd, then $\Phi$ is a daisy, but if $t$ is even, then, depending on $M$, it is possible for $\Phi$ to be either an anemone or a daisy., Comment: 20 pages

We classify all recurrent configurations of the Abelian sandpile model (ASM) on Ferrers graphs. The classification is in terms of decorations of EW-tableaux, which undecorated are in bijection with the minimal recurrent configurations. We introduce decorated permutations, extending to decorated EW-tableaux a bijection between such tableaux and permutations, giving a direct bijection between the decorated permutations and all recurrent configurations of the ASM. We also describe a bijection between the decorated permutations and the intransitive trees of Postnikov, the breadth-first search of which corresponds to a canonical toppling of the corresponding configurations.

In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result of J. Ne\v{s}et\v{r}il and V. R\"{o}dl claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 M. Soki\'{c} proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of S. Solecki and M. Zhao that the class of all finite posets with several linear extensions has the Ramsey property. Using the categorical reinterpretation of the Ramsey property in this paper we prove a common generalization of all these results. We consider multiposets to be structures consisting of several partial orders and several linear orders. We allow partial orders to extend each other in an arbitrary but fixed way, and require that every partial order is extended by at least one of the linear orders. We then show that the class of all finite multiposets conforming to a fixed template has the Ramsey property., Comment: arXiv admin note: text overlap with arXiv:1702.06596

We prove the Schröder case, i.e. the case 〈⋅,e n−d h d 〉, of the conjecture of Haglund et al. (2018) for Δ h m Δ e n−k−1 ′ e n in terms of decorated partially labelled Dyck paths, which we call generalized Delta conjecture. This result extends the Schröder case of the Delta conjecture proved in (D'Adderio, 2017), which in turn generalized the q,t-Schröder in Haglund (2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another combinatorial interpretation of the same polynomial in terms of a new bounce statistic. Moreover, we give two more interpretations of the same polynomial in terms of doubly decorated parallelogram polyominoes, extending some of the results in D'Adderio (2017), which in turn extended results in Aval et al. (2014). Also, we provide combinatorial bijections explaining some of the equivalences among these interpretations., SCOPUS: ar.j, info:eu-repo/semantics/published

An orthogonal representation of a graph is an assignment of nonzero real vectors to its vertices such that distinct non-adjacent vertices are assigned to orthogonal vectors. We prove general lower bounds on the dimension of orthogonal representations of graphs using the Borsuk-Ulam theorem from algebraic topology. Our bounds strengthen the Kneser conjecture, proved by Lov\'asz in 1978, and some of its extensions due to B\'ar\'any, Schrijver, Dol'nikov, and Kriz. As applications, we determine the integrality gap of fractional upper bounds on the Shannon capacity of graphs and the quantum one-round communication complexity of certain promise equality problems., Comment: 18 pages

A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and the in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various subclasses of locally semicomplete digraphs for which we give structural decomposition theorems. As a consequence we obtain several applications, among which an answer to a conjecture of Naserasr and the first and third authors: if an oriented graph is such that the out-neighbourhood of every vertex induces a transitive tournament, then one can partition its vertex set into two acyclic digraphs., Nothing new in this version, which only corrected wrongly typed author names

In this paper we solve two problems of Esperet, Kang and Thomasse as well as Li concerning (i) induced bipartite subgraphs in triangle-free graphs and (ii) van der Waerden numbers. Each time random greedy algorithms allow us to go beyond the Lovasz Local Lemma or alteration method used in previous work, illustrating the power of the algorithmic approach to the probabilistic method., 14 pages; minor edits; to appear in European Journal of Combinatorics

Fix a graph $G$, a list-assignment $L$ for $G$, and $L$-colorings $\alpha$ and $\beta$. An $L$-recoloring sequence, starting from $\alpha$, recolors a single vertex at each step, so that each resulting intermediate coloring is a proper $L$-coloring. An $L$-recoloring sequence transforms $\alpha$ to $\beta$ if its initial coloring is $\alpha$ and its final coloring is $\beta$. We prove there exists an $L$-recoloring sequence that transforms $\alpha$ to $\beta$ and recolors each vertex at most a constant number of times if (i) $G$ is triangle-free and planar and $L$ is a 7-assignment, or (ii) $\mathrm{mad}(G), Comment: 11 pages, 5 figures; to appear in European J. Combinatorics

We study two problems in graph Ramsey theory. In the early 1970's, Erd\H{o}s and O'Neil considered a generalization of Ramsey numbers. Given integers $n,k,s$ and $t$ with $n \ge k \ge s,t \ge 2$, they asked for the least integer $N=f_k(n,s,t)$ such that in any red-blue coloring of the $k$-subsets of $\{1, 2,\ldots, N\}$, there is a set of size $n$ such that either each of its $s$-subsets is contained in some red $k$-subset, or each of its $t$-subsets is contained in some blue $k$-subset. Erd\H{o}s and O'Neil found an exact formula for $f_k(n,s,t)$ when $k\ge s+t-1$. In the arguably more interesting case where $k=s+t-2$, they showed $2^{-\binom{k}{2}}n, Comment: 10 pages

For two graphs $X$ and $Y$ with vertex sets $V(X)$ and $V(Y)$ of the same cardinality $n,$ the friends-and-strangers graph $\mathsf{FS}(X,Y)$ was recently defined by Defant and Kravitz. The vertices of $\mathsf{FS}(X,Y)$ are the bijections from $V(X)$ to $V(Y),$ and two bijections $\sigma$ and $\tau$ are adjacent if they agree everywhere except at two vertices $a,b\in V(X)$ such that $a$ and $b$ are adjacent in $X$ and $\sigma(a)$ and $\sigma(b)$ are adjacent in $Y.$ We study generalized versions of two problems by Alon, Defant, and Kravitz. First, we show that if $X$ and $Y$ have minimum degrees $\delta(X)$ and $\delta(Y)$ that satisfy $\delta(X)> n/2, \delta(Y)>n/2,$ and $2\min(\delta(X), \delta(Y))+3\max(\delta(X), \delta(Y))\ge 3n,$ then $\mathsf{FS}(X,Y)$ is connected. As a corollary, we settle a recent conjecture by Alon, Defant, and Kravitz stating that there exists a number $d_n = 3n/5 + O(1)$ such that if both $X$ and $Y$ have minimum degrees at least $d_n,$ the graph $\mathsf{FS}(X,Y)$ is connected. When $X$ and $Y$ are bipartite, a parity obstruction prevents $\mathsf{FS}(X,Y)$ from being connected. We show that if $X$ and $Y$ are edge-subgraphs of $K_{r,r}$ that satisfy $\delta(X)+\delta(Y)\ge 3r/2+1,$ then the graph $\mathsf{FS}(X,Y)$ has exactly two connected components. As a corollary, we provide an almost complete answer to another recent question of Alon, Defant, and Kravitz asking for the minimum number $d^*_{r,r}$ such that for any edge-subgraph $X$ of $K_{r,r}$ satisfying $\delta(X)\ge d^*_{r,r},$ the graph $\mathsf{FS}(X,K_{r,r})$ has exactly two connected components. We show that $d^*_{r,r} = r/2+1$ when $r$ is even and $d^*_{r,r}\in \{\lceil r/2\rceil, \lceil r/2\rceil+1\}$ when $r$ is odd., Comment: 28 pages, 16 figures

A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two $k$-colorings are $k$-equivalent if we can transform one into the other using Kempe swaps. The triangulated toroidal grid, $T[m\times n]$, is formed from (a toroidal embedding of) the Cartesian product of $C_m$ and $C_n$ by adding parallel diagonals inside all 4-faces. Mohar and Salas showed that not all 4-colorings of $T[m\times n]$ are 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of $T[m\times n]$ are 6-equivalent. They asked whether the same is true for 5-colorings. We answer their question affirmatively when $m,n\ge 6$. Further, we show that if $G$ is 6-regular with a toroidal embedding where every non-contractible cycle has length at least 7, then all 5-colorings of $G$ are 5-equivalent. Our results relate to the antiferromagnetic Pott's model in statistical mechanics., Comment: 17 pages, 16 figures; 4th version incorporates minor referee feedback; to appear in European Journal of Combinatorics

We study the family of \emph{derangement action digraphs}, which are a subfamily of the group action graphs introduced in [Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg, Group action graphs and parallel architectures, \emph{SIAM J. Comput.} 19 (1990), no. 3, 544--569]. For any non-empty set $X$ and a non-empty subset $S$ of $\Der(X)$, the set of derangments of $X$, we define the derangement action digraph $\rm\overrightarrow{DA}(X;S)$ to have vertex set $X$, and an arc from $x$ to $y$ if and only if $y=x^s$ for some $s\in S$. In common with Cayley graphs and digraphs, derangement action digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on $S$ under which $\rm\overrightarrow{DA}(X;S)$ may be viewed as a simple graph of valency $|S|$, and we call such graphs derangement action graphs. Also we investigate the structural and symmetry properties of these digraphs and graphs. Several open problems are posed and many examples are given., Comment: 15 pages, 1 figure

Every graph G can be embedded in a Euclidean space as a two-distance set. The Euclidean representation number of G is the smallest dimension in which G is representable by such an embedding. We consider spherical and J-spherical representation numbers of G and give exact formulas for these numbers using multiplicities of polynomials that are defined by the Caley-Menger determinant. One of the main results of the paper are explicit formulas for the representation numbers of the join of graphs which are obtained from W. Kuperberg's type theorem for two-distance sets., 20 pages

We show that a weaker version of the well-known theorem of Morlaye and Joly on diagonal equations is a simple consequence of a restricted variable version of the Chevalley-Warning theorem. Moreover, we extend the result of Morlaye and Joly to the case of an equation of the form $$b_1D_{m_1}(X_1,a_1)+\dots+b_nD_{m_n}(X_n,a_n)=c,$$ where $D_{m_1}(X_1,a_1),\dots,D_{m_n}(X_n,a_n)$ are Dickson polynomials., 7 pages; Example 1 and reference [5] added

A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. We study the structure of groups acting regularly on the point set of a generalized quadrangle. In particular, we provide a characterization of the generalized quadrangles with a group of automorphisms acting regularly on both the point set and the line set and show that such a thick generalized quadrangle does not admit a polarity. Moreover, we prove that a group $G$ acting regularly on the point set of a generalized quadrangle of order $(u^2, u^3)$ or $(s,s)$, where $s$ is odd and $s+1$ is coprime to $3$, cannot have any nonabelian minimal normal subgroups., Comment: 16 pages; to appear in European Journal of Combinatorics

With a main tool is signed graphs, we give a full description of the characteristic quasi-polynomials of ideals of classical root systems ($ABCD$) with respect to the integer and root lattices. As a result, we obtain a full description of the characteristic polynomials of the toric arrangements defined by these ideals. As an application, we provide a combinatorial verification to the fact that the characteristic polynomial of every ideal subarrangement factors over the dual partition of the ideal in the classical cases., Comment: 17 pages, we welcome comments

Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid $M$ is Hamiltonian if it has a spanning circuit. A matroid $M$ is nested if and only if its Hamiltonian flats form a chain under inclusion; $M$ is laminar if and only if, for every $1$-element independent set $X$, the Hamiltonian flats of $M$ containing $X$ form a chain under inclusion. We generalize these notions to define the classes of $k$-closure-laminar and $k$-laminar matroids. This paper focuses on structural properties of these classes noting that, while the second class is always minor-closed, the first is if and only if $k \le 3$. The main results are excluded-minor characterizations for the classes of 2-laminar and 2-closure-laminar matroids., 12 pages

By a theorem of Drisko, any $2n-1$ matchings of size $n$ in a bipartite graph have a partial rainbow matching of size $n$. Inspired by discussion of Bar\'at, Gy\'arf\'as and S\'ark\"ozy, we conjecture that if $n$ is odd then the same is true also in general graphs, and that if $n$ is even then $2n$ matchings of size $n$ suffice. We prove that any $3n-2$ matchings of size $n$ have a partial rainbow matching of size $n$., Comment: 6 pages

We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph $G$ on $n$ vertices has a rainbow cycle on at least $n - O(n^{3/4})$ vertices, by showing that $G$ has a rainbow cycle on at least $n - O(\log n \sqrt{n})$ vertices. Second, by modifying the argument of Hatami and Shor which gives a lower bound for the length of a partial transversal in a Latin Square, we prove that every properly colored complete graph has a Hamilton cycle in which at least $n - O((\log n)^2)$ different colors appear. For large $n$, this is an improvement of the previous best known lower bound of $n - \sqrt{2n}$ of Andersen., Comment: 12 pages

How large an antichain can we find inside a given downset in the lattice of subsets of [n]? Sperner's theorem asserts that the largest antichain in the whole lattice has size the binomial coefficient C(n, n/2); what happens for general downsets? Our main results are a Dilworth-type decomposition theorem for downsets, and a new proof of a result of Engel and Leck that determines the largest possible antichain size over all downsets of a given size. We also prove some related results, such as determining the maximum size of an antichain inside the downset that we conjecture minimizes this quantity among downsets of a given size., 18 pages, 3 figures

Recently, Nunge studied Eulerian polynomials on segmented permutations, namely \emph{generalized Eulerian polynomials}, and further asked whether their coefficients form unimodal sequences. In this paper, we prove the stability of the generalized Eulerian polynomials and hence confirm Nunge's conjecture. Our proof is based on Br\"and\'en's stable multivariate Eulerian polynomials. By acting on Br\"and\'en's polynomials with a stability-preserving linear operator, we get a multivariate refinement of the generalized Eulerian polynomials. To prove Nunge's conjecture, we also develop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials., Comment: to appear in European Journal of Combinatorics

A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting if each set in $\mathcal{A}$ $t$-intersects each set in $\mathcal{B}$. A family $\mathcal{H}$ is hereditary if for each set $A$ in $\mathcal{H}$, all the subsets of $A$ are in $\mathcal{H}$. The $r$th level of $\mathcal{H}$, denoted by $\mathcal{H}^{(r)}$, is the family of $r$-element sets in $\mathcal{H}$. A set $B$ in $\mathcal{H}$ is a base of $\mathcal{H}$ if for each set $A$ in $\mathcal{H}$, $B$ is not a proper subset of $A$. Let $��(\mathcal{H})$ denote the size of a smallest base of $\mathcal{H}$. We show that for any integers $t$, $r$, and $s$ with $1 \leq t \leq r \leq s$, there exists an integer $c(r,s,t)$ such that the following holds for any hereditary family $\mathcal{H}$ with $��(\mathcal{H}) \geq c(r,s,t)$. If $\mathcal{A}$ is a non-empty subfamily of $\mathcal{H}^{(r)}$, $\mathcal{B}$ is a non-empty subfamily of $\mathcal{H}^{(s)}$, $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting, and $|\mathcal{A}| + |\mathcal{B}|$ is maximum under the given conditions, then for some set $I$ in $\mathcal{H}$ with $t \leq |I| \leq r$, either $\mathcal{A} = \{A \in \mathcal{H}^{(r)} \colon I \subseteq A\}$ and $\mathcal{B} = \{B \in \mathcal{H}^{(s)} \colon |B \cap I| \geq t\}$, or $r = s$, $t < |I|$, $\mathcal{A} = \{A \in \mathcal{H}^{(r)} \colon |A \cap I| \geq t\}$, and $\mathcal{B} = \{B \in \mathcal{H}^{(s)} \colon I \subseteq B\}$. This was conjectured by the author for $t=1$ and generalizes well-known results for the case where $\mathcal{H}$ is a power set., 15 pages. arXiv admin note: text overlap with arXiv:1805.05241

Baker and Norine proved a Riemann--Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Bj\"orner, Lov\'asz and Shor. We use this connection to prove Riemann--Roch-type results on directed graphs. We give a simple proof for a Riemann--Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann--Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework., Comment: 22 pages, 4 figures

Komlos [Tiling Turan theorems, Combinatorica, 20,2 (2000), 203{218] determined the asymptotically optimal minimum degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph. We show that the minimum degree condition can be relaxed in the sense that we require only a given fraction of vertices to have the prescribed degree., Comment: 13 pages, 1 figure, accepted to the European Journal of Combinatorics

A detailed description of the structure of two-ended arc-transitive digraphs is given. It is also shown that several sets of conditions, involving such concepts as Property Z, local quasi-primitivity and prime out-valency, imply that an arc-transitive digraph must be highly-arc-transitive. These are then applied to give a complete classification of two-ended highly-arc-transitive digraphs with prime in- and out-valencies., To appear in European Journal of Combinatorics. Statements of Corollaries 14, 16 and 17 corrected