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We introduce coloring groups, which are permutation groups obtained from a proper edge coloring of a graph. These groups generalize the generalized toggle groups of Striker (which themselves generalize the toggle groups introduced by Cameron and Fon-der-Flaass). We present some general results connecting the structure of a coloring group to the structure of its graph coloring, providing graph-theoretic characterizations of the centralizer and primitivity of a coloring group. We apply these results particularly to generalized toggle groups arising from trees as well as coloring groups arising from the independence posets introduced by Thomas and Williams.

A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring such that no two vertices in the same edge have the same color, and each $k$-element subset of colors appears on at most one edge. The harmonious number $h(H)$ is the least number of colors needed for such a coloring. The paper contains a new proof of the upper bound $h(H)=O(\sqrt[k]{k!m})$ on the harmonious number of hypergraphs of maximum degree $\Delta$ with $m$ edges. We use the local cut lemma of A. Bernshteyn.

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been further extended to binary structures, in several (basically equivalent) ways. We prove that a class of binary relational structures (that is: edge-colored partially directed graphs) has bounded twin-width if and only if it is a first-order transduction of a~proper permutation class. As a by-product, we show that every class with bounded twin-width contains at most $2^{O(n)}$ pairwise non-isomorphic $n$-vertex graphs.

An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. Carvalho, Lucchesi, and Murty showed that every brick $G$ different from $K_4$ and $\overline{C_6}$ has at least $\Delta-2$ removable edges, where $\Delta$ is the maximum degree of $G$. In this paper, we generalize the result to irreducible near-bricks, where a graph is irreducible if it contains no single ear of length three or more.

We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if $G=C^d_{2k}$ is the $d$th power of the cartesian product of the even cycle $C_{2k}$ with itself, and at most $2d-1$ edges of $G$ are precolored, then there is a proper $2d$-edge coloring of $G$ that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least $4$ from each other. For odd cycles of length at least $5$, we prove that if $G=C^d_{2k+1}$ is the $d$th power of the cartesian product of the odd cycle $C_{2k+1}$ with itself ($k\geq2$), and at most $2d$ edges of $G$ are precolored, then there is a proper $(2d+1)$-edge coloring of $G$ that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410--444].

In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian: {\it Let $D$ be a 2-strong digraph of order $n\geq 9$. If $n-1$ vertices of $D$ have degrees at least $n+k$ and the remaining vertex has degree at least $n-k-4$, where $k$ is a non-negative integer, then $D$ is Hamiltonian}. This is an extension of Ghouila-Houri's theorem for 2-strong digraphs and is a generalization of an early result of the author (DAN Arm. SSR (91(2):6-8, 1990). The obtained result is best possible in the sense that for $k=0$ there is a digraph of order $n=8$ (respectively, $n=9$) with the minimum degree $n-4=4$ (respectively, with the minimum $n-5=4$) whose $n-1$ vertices have degrees at least $n-1$, but it is not Hamiltonian. We also give a new sufficient condition for a 3-strong digraph to be Hamiltonian-connected.

A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected graphs and obtain a more detailed result that relates the number of vertices to the operations involved in constructing a respective uniformly $3$-connected graph. Furthermore, we investigate how crossing numbers and treewidths behave under the mentioned constructions. We demonstrate how these results can be utilized to study the structure and properties of uniformly $3$-connected graphs with minimum number of vertices of minimum degree.

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, the authors of a recent paper showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$. This exhibited the fact that random permutations pack consecutive patterns near-perfectly. We use entirely different methods, namely the Stein-Chen method of Poisson approximation, to reprove and slightly improve their result.

This paper is devoted to the study of particular geometrically defined intersection classes of graphs. Those were previously studied by Magnant and Martin, who proved that these graphs have arbitrary large chromatic number, while being triangle-free. We give several structural properties of these graphs, and we raise several questions.

A walk $u_0u_1 \ldots u_{k-1}u_k$ is a \textit{weakly toll walk} if $u_0u_i \in E(G)$ implies $u_i = u_1$ and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. A set $S$ of vertices of $G$ is {\it weakly toll convex} if for any two non-adjacent vertices $x,y \in S$ any vertex in a weakly toll walk between $x$ and $y$ is also in $S$. The {\em weakly toll convexity} is the graph convexity space defined over weakly toll convex sets. Many studies are devoted to determine if a graph equipped with a convexity space is a {\em convex geometry}. An \emph{extreme vertex} is an element $x$ of a convex set $S$ such that the set $S\backslash\{x\}$ is also convex. A graph convexity space is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, $m^3$, and toll convexities, respectively. Other important classes of graphs can also be characterized in this way. In this paper, we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity.

We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\Psi$, in the language of two total orders, the probability $p_{n,\Psi}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $\Psi$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,\Psi}$: (i) it is either bounded away from $0$, or decays exponentially fast; (ii) the set of possible limits is dense in $[0,1]$. Our tools come mainly from analytic combinatorics and singularity analysis.

We study the Maker-Breaker domination game played by Dominator and Staller on the vertex set of a given graph. Dominator wins when the vertices he has claimed form a dominating set of the graph. Staller wins if she makes it impossible for Dominator to win, or equivalently, she is able to claim some vertex and all its neighbours. Maker-Breaker domination number $\gamma_{MB}(G)$ ($\gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of moves for Dominator to guarantee his winning when he plays first (second). We investigate these two invariants for the Cartesian product of any two graphs. We obtain upper bounds for the Maker-Breaker domination number of the Cartesian product of two arbitrary graphs. Also, we give upper bounds for the Maker-Breaker domination number of the Cartesian product of the complete graph with two vertices and an arbitrary graph. Most importantly, we prove that $\gamma'_{MB}(P_2\square P_n)=n$ for $n\geq 1$, $\gamma_{MB}(P_2\square P_n)$ equals $n$, $n-1$, $n-2$, for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively. For the disjoint union of $P_2\square P_n$s, we show that $\gamma_{MB}'(\dot\cup_{i=1}^k(P_2\square P_n)_i)=k\cdot n$ ($n\geq 1$), and that $\gamma_{MB}(\dot\cup_{i=1}^k(P_2\square P_n)_i)$ equals $k\cdot n$, $k\cdot n-1$, $k\cdot n-2$ for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq 13$, respectively.

The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours the vertices in the order of $\sigma$ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly $\chi(G)$ colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only $\chi(G)$ colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no $K_4$-minor free graph is ugly. Moreover, our proofs are constructive, and imply the existence of polynomial-time algorithms to compute good connected orderings for these graph classes.

We explore new connections between complete non-ambiguous trees (CNATs) and permutations. We give a bijection between tree-like tableaux and a specific subset of CNATs. This map is used to establish and solve a recurrence relation for the number of tree-like tableaux of a fixed size without occupied corners, proving a conjecture by Laborde-Zubieta. We end by establishing a row/column swapping operation on CNATs and identify new areas for future research.

In this corrigendum, we give a counterexample to Theorem 5.2 in "On the monophonic rank of a graph" [Discrete Math. Theor. Comput. Sci. 24:2 (2022) #3]. We also present a polynomial-time algorithm for computing the monophonic rank of a starlike graph.

This paper studies the relationship between the modified Foata$\unicode{x2013}$Strehl action (a.k.a. valley-hopping)$\unicode{x2014}$a group action on permutations used to demonstrate the $\gamma$-positivity of the Eulerian polynomials$\unicode{x2014}$and the number of rixed points $\operatorname{rix}$$\unicode{x2014}$a recursively-defined permutation statistic introduced by Lin in the context of an equidistribution problem. We give a linear-time iterative algorithm for computing the set of rixed points, and prove that the $\operatorname{rix}$ statistic is homomesic under valley-hopping. We also demonstrate that a bijection $\Phi$ introduced by Lin and Zeng in the study of the $\operatorname{rix}$ statistic sends orbits of the valley-hopping action to orbits of a cyclic version of valley-hopping, which implies that the number of fixed points $\operatorname{fix}$ is homomesic under cyclic valley-hopping.

For the given bipartite graphs $G_1,G_2,\ldots,G_t$, the multicolor bipartite Ramsey number $BR(G_1,G_2,\ldots,G_t)$ is the smallest positive integer $b$ such that any $t$-edge-coloring of $K_{b,b}$ contains a monochromatic subgraph isomorphic to $G_i$, colored with the $i$th color for some $1\leq i\leq t$. We compute the exact values of the bipartite Ramsey numbers $BR(C_8,C_{2n})$ for $n\geq2$.

In a recent preprint, Matherne, Morales and Selover conjectured that two different representations of unit interval posets are related by the famous zeta map in $q,t$-Catalan combinatorics. This conjecture was proved recently by G\'elinas, Segovia and Thomas using induction. In this short note, we provide a bijective proof of the same conjecture with a reformulation of the zeta map using left-aligned colored trees, first proposed in the study of parabolic Tamari lattices.

Inspired by the split decomposition of graphs and rank-width, we introduce the notion of $r$-splits. We focus on the family of $r$-splits of a graph of order $n$, and we prove that it forms a hypergraph with several properties. We prove that such hypergraphs can be represented using only $\mathcal O(n^{r+1})$ of its hyperedges, despite its potentially exponential number of hyperedges. We also prove that there exist hypergraphs that need at least $\Omega(n^r)$ hyperedges to be represented, using a generalization of set orthogonality.

Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs. For an Erd\H{o}s-Renyi random graph, we identify a threshold probability at which with high probability the symmetric edge polytope shares many facet-supporting hyperplanes with that of a complete graph. We also investigate the relationship between the average local clustering, also known as the Watts-Strogatz clustering coefficient, and the number of facets for graphs with either a fixed number of edges or a fixed degree sequence. We use well-known Markov Chain Monte Carlo sampling methods to generate empirical evidence that for a fixed degree sequence, higher average local clustering in a connected graph corresponds to higher facet numbers in the associated symmetric edge polytope.

Let c be an integer. A c-partite tournament is an orientation of a complete c-partite graph. A c-partite tournament is rich if it is strong, and each partite set has at least two vertices. In 1996, Guo and Volkmann characterized the structure of all rich c-partite tournaments without (c + 1)-cycles, which solved a problem by Bondy. They also put forward a problem that what the structure of rich c-partite tournaments without (c + k)-cycles for some k>1 is. In this paper, we answer the question of Guo and Volkmann for k = 2.

The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.

Mixed graphs have both directed and undirected edges. A mixed cage is a regular mixed graph of given girth with minimum possible order. In this paper mixed cages are studied. Upper bounds are obtained by general construction methods and computer searches.

We establish exact values for the $2$-limited broadcast domination number of various grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ and all $n\geq m$, $P_m \square C_3$ for all $m \geq 3$, and $P_m \square C_n$ for $4\leq m \leq 5$ and all $n \geq m$. We also produce periodically optimal values for $P_m \square C_4$ and $P_m \square C_6$ for $m \geq 3$, $P_4 \square P_n$ for $n \geq 4$, and $P_5 \square P_n$ for $n \geq 5$. Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.

We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more unified manner, and provide a complete answer to one of his previously unresolved questions. We consider finding words with specific pseudoperiod and having the smallest possible critical exponent. Finally, we consider the problem of determining whether a finite word is pseudoperiodic of a given size, and show that it is NP-complete.

Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete.

Consider the following hat guessing game. A bear sits on each vertex of a graph $G$, and a demon puts on each bear a hat colored by one of $h$ colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess $g$ colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement. We introduce a new parameter - fractional hat chromatic number $\hat{\mu}$, arising from the hat guessing game. The parameter $\hat{\mu}$ is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of $G$, and to compute the exact value of $\hat{\mu}$ of cliques, paths, and cycles.

The (bitwise) complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An $\textit{antisquare}$ is a nonempty word of the form $x\, \overline{x}$. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is $(5+\sqrt{5})/2$. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is $\textit{good}$ if the only antisquares it contains are $01$ and $10$. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length $n$ and determine the repetition threshold between polynomial and exponential growth for the number of good words.

In the Maker-Breaker domination game played on a graph $G$, Dominator's goal is to select a dominating set and Staller's goal is to claim a closed neighborhood of some vertex. We study the cases when Staller can win the game. If Dominator (resp., Staller) starts the game, then $\gamma_{\rm SMB}(G)$ (resp., $\gamma_{\rm SMB}'(G)$) denotes the minimum number of moves Staller needs to win. For every positive integer $k$, trees $T$ with $\gamma_{\rm SMB}'(T)=k$ are characterized and a general upper bound on $\gamma_{\rm SMB}'$ is proved. Let $S = S(n_1,\dots, n_\ell)$ be the subdivided star obtained from the star with $\ell$ edges by subdividing its edges $n_1-1, \ldots, n_\ell-1$ times, respectively. Then $\gamma_{\rm SMB}'(S)$ is determined in all the cases except when $\ell\ge 4$ and each $n_i$ is even. The simplest formula is obtained when there are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two smallest such numbers, then $\gamma_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil \log_2(n_1+n_2+1)\rceil$. For caterpillars, exact formulas for $\gamma_{\rm SMB}$ and for $\gamma_{\rm SMB}'$ are established.

Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021). In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.

In this work we study line arrangements consisting in lines passing through three non-aligned points. We call them triangular arrangements. We prove that any combinatorics of a triangular arrangement is always realized by a Roots-of-Unity-Arrangement, which is a particular class of triangular arrangements. Among these Roots-of Unity-Arrangements, we provide conditions that ensure their freeness. Finally, we give two triangular arrangements having the same weak combinatorics, such that one is free but the other one is not.

Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle.

This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $\gamma_{R}(K_{n,k})$ and total Roman domination number $\gamma_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $\gamma_{R}(K_{n,k}) =\gamma_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $\gamma_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature.

Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were defined in a purely combinatorial way. This led to extensions of earlier results about points and halfplanes to pseudohalfplanes, including polychromatic colorings and discrete Helly-type theorems about pseudohalfplanes. Here we continue this line of research and introduce the notion of convex sets of such pseudohalfplane hypergraphs. In this context we prove several results corresponding to classical results about convexity, namely Helly's Theorem, Carath\'eodory's Theorem, Kirchberger's Theorem, Separation Theorem, Radon's Theorem and the Cup-Cap Theorem. These results imply the respective results about pseudoconvex sets in the plane defined using pseudohalfplanes. It turns out that most of our results can be also proved using oriented matroids and topological affine planes (TAPs) but our approach is different from both of them. Compared to oriented matroids, our theory is based on a linear ordering of the vertex set which makes our definitions and proofs quite different and perhaps more elementary. Compared to TAPs, which are continuous objects, our proofs are purely combinatorial and again quite different in flavor. Altogether, we believe that our new approach can further our understanding of these fundamental convexity results.

We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a characterization of the set of possible limits and its structure as a metric space. For random trees the subtree size topology arises in the context of algorithms for searching and sorting when applied to random input, resulting in a sequence of nested trees. For these we obtain a structural result based on a local version of exchangeability. This in turn leads to a central limit theorem, with possibly mixed asymptotic normality.