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Suppose $G$ is a graph and $L$ is a list assignment for $G$. A request of $L$ is a function $r$ with nonempty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$. The triple $(G,L,r)$ is $\epsilon$-satisfiable if there exists a proper $L$-coloring $f$ of $G$ such that $f(v) = r(v)$ for at least $\epsilon|D|$ vertices in $D$. We say $G$ is $(k, \epsilon)$-flexible if $(G,L',r')$ is $\epsilon$-satisfiable whenever $L'$ is a $k$-assignment for $G$ and $r'$ is a request of $L'$. It is known that a graph $G$ is not $(k, \epsilon)$-flexible for any $k$ if and only if $\epsilon > 1/ \rho(G)$ where $\rho(G)$ is the Hall ratio of $G$. The list flexibility number of a graph $G$, denoted $\chi_{\ell flex}(G)$, is the smallest $k$ such that $G$ is $(k,1/ \rho(G))$-flexible. A fundamental open question on list flexibility numbers asks: Is there a graph with list flexibility number greater than its coloring number? In this paper, we show that the list flexibility number of any complete multipartite graph $G$ is at most the coloring number of $G$. With this as a starting point, we study list epsilon flexibility functions of complete bipartite graphs which was first suggested by Kaul, Mathew, Mudrock, and Pelsmajer in 2024. Specifically, we completely determine the list epsilon flexibility function of $K_{m,n}$ when $m \in \{1,2\}$ and establish additional bounds. Our proofs reveal a connection to list coloring complete bipartite graphs with asymmetric list sizes which is a topic that was explored by Alon, Cambie, and Kang in 2021., Comment: 18 pages

A multigraph in which adjacent vertices have different degrees is called locally irregular. The locally irregular edge coloring is an edge coloring of a multigraph $G$ in which every color induces a locally irregular submultigraph of $G$. We denote by $\operatorname{lir}(G)$ the locally irregular chromatic index of a multigraph $G$, which is the smallest number of colors required in a locally irregular edge coloring of $G$, given that such a coloring of $G$ exists. By $^2G$ we denote a 2-multigraph obtained from a simple graph $G$ by doubling each its edge. In 2022 Grzelec and Wo\'zniak conjectured that $\operatorname{lir}(^2G) \leq 2$ for every connected simple graph $G$ different from $K_2$; the conjecture is known as Local Irregularity Conjecture for 2-multigraphs. In this paper, we prove this conjecture in the case of regular graphs, split graphs, and some particular families of subcubic graphs. Moreover, we provide a constant upper bound on the locally irregular chromatic index of planar 2-multigraphs (except for $^2K_2$), and we obtain a better constant upper bound on $\operatorname{lir}(^2G)$ if $G$ is a simple subcubic graph different from $K_2$. In the proofs, special decompositions of graphs and the relation of Local Irregularity Conjecture to the well-known 1-2-3 Conjecture are utilized., Comment: 22 pages, 11 figures

A coloring of the vertex set of a graph using the colors red and blue is a closed neighborhood balanced coloring if for each vertex there are an equal number of red and blue vertices in its closed neighborhood. A graph with such a coloring is called a CNBC graph. Freyberg and Marr studied the related class of NBC graphs where closed neighborhood is replaced by open neighborhood. We prove results about CNBC graphs and NBC graphs. We show that the class of CNBC graphs is not hereditary, that the sizes of the color classes can be arbitrarily different, and that if the sizes of the color classes are equal, then a graph is a CNBC graph if and only if its complement is an NBC graph. When the sizes of the color classes are equal, we show that the join of two CNBC graphs is a CNBC graph, and the lexicographic product of a CNBC graph with any graph is a CNBC graph. We prove that the Cartesian product of any CNBC graph and any NBC graph is a CNBC graph, and characterize when a hypercube is an NBC graph or a CNBC graph, but show that the product of two CNBC graphs need not be an NBC graph. We show that the strong product of a CNBC graph with any graph is a CNBC graph. We construct infinite families of circulants that are CNBC graphs, and give characterizations of CNBC trees, generalized Petersen graphs, cubic circulants and quintic circulants when $n\equiv 2 \pmod 4$., Comment: 22 pages, 11 figures, 4 tables

We give a pen and paper and (comparatively) much simpler proof to verify of the Four Colour Theorem., Comment: 13 pages; this is a follow-up to my first submission on this problem (though the rights are not the same) and which I firmly consider to be my last attempt. Acknowledgments will follow (officially) should the proof be approved

Bonamy et al. (2023) proved that an optimal edge coloring of a simple triangle--free graph $G$ can be reached from any given proper edge coloring of $G$ through a series of Kempe changes. We show that a small modification of their proof gives a possibility to obtain a similar result for a larger class of simple graphs consisting of all triangle-free and all chordless graphs (a graph $G$ is chordless if in every cycle $C$ of $G$ any two nonconsecutive vertices of $C$ are not adjacent)., Comment: 9 pages

We study the paintability, an on-line version of choosability, of complete multipartite graphs. We do this by considering an equivalent chip game introduced by Duraj, Gutowski, and Kozik. We consider complete multipartite graphs with $ n $ parts of size at most 3. Using a computational approach, we establish upper bounds on the paintability of such graphs for small values of $ n. $ The choosability of complete multipartite graphs is closely related to value $ p(n, m) $, the minimum number of edges in a $n$-uniform hypergraph with no panchromatic $m$-coloring. We consider an online variant of this parameter $ p_{OL}(n, m), $ introduced by Khuzieva et al. using a symmetric chip game. With this symmetric chip game, we find an improved upper bound for $ p_{OL}(n, m)$ when $m \geq 3$ and $n$ is large. Our method also implies a lower bound on the paintability of complete multipartite graphs with $m \geq 3$ parts of equal size., Comment: This paper originates from an Illinois Math Lab undergraduate project at University of Illinois Urbana-Champaign

We present a necessary and sufficient condition for determining whether a hypergraph can be embedded in a high-dimensional space, and establish an upper bound for the chromatic number of such hypergraph, which can be viewed as the high-dimensional extension of the Heawood conjecture and Wagner's Theorem, respectively. Based on them, for a graph of order $n$, there exists accordingly an algorithm with a complexity of $O(n\log^2 n)$ that can determine the upper bound for the chromatic number. Specifically, we prove: (i) The graph-closure of a $d$-uniform-topological hypergraph is a closed-$R^d$-graph if and only if its minors include neither $K_{d+3}^d$ nor $K_{3,d+1}^d$; (ii) Let $G$ be a special triangulated $R^d$-graph, $G^v$ be the vertex-hypergraph of $G$, then $G^v$ is $3\cdot 2^{d-1}$-choosable., Comment: 19 pages, 12 figures

An edge-coloring of a graph $G$ assigns a color to each edge of $G$. An edge-coloring is a parity edge-coloring if for each path $P$ in $G$, it uses some color on an odd number of edges in $P$. It is a strong parity edge-coloring if for every open walk $W$ in $G$, it uses some color an odd number of times along $W$. The minimum numbers of colors in parity and strong parity edge-colorings of $G$ are denoted $p(G)$ and $\hat{p}(G)$, respectively. We characterize strong parity edge-colorings and use this characterization to prove lower bounds on $\hat{p}(G)$ and answer several questions of Bunde, Milans, West, and Wu. The applications are as follows. (1) We prove the conjecture that $\hat{p}(K_{s,t})=s \circ t$, where $s \circ t$ is the Hopf-Stiefel function. (2) We show that $\hat{p}(G)$ for a connected $n$-vertex graph $G$ equals the known lower bound $\lceil \log_2 n \rceil$ if and only if $G$ is a subgraph of the hypercube $Q_{\lceil \log_2 n \rceil }$. (3) We asymptotically compute $\hat{p}(G)$ when $G$ is the $\ell$th distance-power of a path, proving $\hat{p}(P_n^\ell)\sim\ell \lceil {\log_2 n} \rceil$. (4) We disprove the conjecture that $\hat{p}(G)=p(G)$ when $G$ is bipartite by constructing bipartite graphs $G$ such that $\hat{p}(G)/p(G)$ is arbitrarily large; in particular, with $\hat{p}(G)\ge\frac{1-o(1)}3 k\ln k$ and $p(G)\le2k+k^{1/3}$., Comment: 17 pages

Recently, Van Hoeve proposed an algorithm for graph coloring based on an integer flow formulation on decision diagrams for stable sets. We prove that the solution to the linear flow relaxation on exact decision diagrams determines the fractional chromatic number of a graph. This settles the question whether the decision diagram formulation or the fractional chromatic number establishes a stronger lower bound. It also establishes that the integrality gap of the linear programming relaxation is O(log n), where n represents the number of vertices in the graph. We also conduct experiments using exact decision diagrams and could determine the chromatic number of r1000.1c from the DIMACS benchmark set. It was previously unknown and is one of the few newly solved DIMACS instances in the last 10 years., Comment: 11 pages, 1 Figure, 1 Table, source code published at https://github.com/trewes/ddcolors, Scripts, logfiles, results and instructions for the experiments are archived at https://bonndata.uni-bonn.de/dataset.xhtml?persistentId=doi:10.60507/FK2/ZE9C3L

The star chromatic index of a graph $G$, denoted by $\chi^\prime_s(G)$, is the smallest integer $k$ for which $G$ admits a proper edge coloring with $k$ colors such that every path and cycle of length four is not bicolored. Let $d$ be the greatest common divisor of $n$ and $k$. Zhu~et~al. (\footnotesize{Discussiones Mathematicae: Graph Theory, 41(2): 1265, 2021}) showed that for every integers $k$ and $n> 2k$ with $d\geq 3$, generalized Petersen graph $GP(n,k)$ admits a 5-star edge coloring, with the exception of the case that $d = 3$, $k\neq d$ and $\frac{n}{3}= 1\pmod{3}$. Also, they conjectured that for every $n>2k$, $\chi^\prime_s(GP(n,k))\leq 5$, except $GP(3,1)$. In this paper, we prove that for every $GP(n,k)$ with $n\geq 2k$ and $d\geq 3$ their conjecture is true. In fact, we provide a 5-star edge coloring of $GP(n,k)$, where $n\geq 2k$ and $d\geq 3$. We also obtain some results for 5-star edge coloring of $GP(n,k)$ with $d=2$. Moreover, Dvo{\v{r}}{\'a}k et al. ({\footnotesize Journal of Graph Theory, 72(3):313-326, 2013}) conjectured that the star of chromatic index of subcubic graphs is at most 6. Thus, our results also prove this conjecture for the generalized Petersen graphs, as a class of subcubic graphs.

An edge-colouring of a graph $G$ is said to be colour-balanced if there are equally many edges of each available colour. We are interested in finding a colour-balanced perfect matching within a colour-balanced clique $K_{2nk}$ with a palette of $k$ colours. While it is not necessarily possible to find such a perfect matching, one can ask for a perfect matching as close to colour-balanced as possible. In particular, for a colouring $c:E(K_{2nk})\rightarrow [k]$, we seek to find a perfect matching $M$ minimising $f(M) = \sum_{i=1}^k\bigl||c^{-1}(i)\cap M|-n\bigr|$. The previous best upper bound, due to Pardey and Rautenbach, was $\min f(M)\leq \mathcal{O}(k\sqrt{nk\log k})$. We remove the $n$-dependence, proving the existence of a matching $M$ with $f(M)\leq 4^{k^2}$ for all $k$., Comment: 12 pages

For any positive integer $k$, the reconfiguration graph for all $k$-colorings of a graph $G$, denoted by $\mathcal{R}_k(G)$, is the graph where vertices represent the $k$-colorings of $G$, and two $k$-colorings are joined by an edge if they differ in color on exactly one vertex. Bonamy et al. established that for any $2$-chromatic $P_5$-free graph $G$, $\mathcal{R}_k(G)$ is connected for each $k\geq 3$. On the other hand, Feghali and Merkel proved the existence of a $7p$-chromatic $P_5$-free graph $G$ for every positive integer $p$, such that $\mathcal{R}_{8p}(G)$ is disconnected. In this paper, we offer a detailed classification of the connectivity of $\mathcal{R} _k(G) $ concerning $t$-chromatic $P_5$-free graphs $G$ for cases $t=3$, and $t\geq4$ with $t+1\leq k \leq {t\choose2}$. We demonstrate that $\mathcal{R}_k(G)$ remains connected for each $3$-chromatic $P_5$-free graph $G$ and each $k \geq 4$. Furthermore, for each $t\geq4$ and $t+1 \leq k \leq {t\choose2}$, we provide a construction of a $t$-chromatic $P_5$-free graph $G$ with $\mathcal{R}_k(G)$ being disconnected. This resolves a question posed by Feghali and Merkel.

The distinguishing index gives a measure of symmetry in a graph. Given a graph $G$ with no $K_2$ component, a distinguishing edge coloring is a coloring of the edges of $G$ such that no non-trivial automorphism preserves the edge coloring. The distinguishing index, denoted $\operatorname{Dist^{\prime}}(G)$, is the smallest number of colors needed for a distinguishing edge coloring. The Mycielskian of a graph $G$, denoted $\mu(G)$, is an extension of $G$ introduced by Mycielski in 1955. In 2020, Alikhani and Soltani conjectured a relationship between $operatorname{Dist^{\prime}}(G)$ and $operatorname{Dist^{\prime}}(\mu(G))$. We prove that for all graphs $G$ with at least 3 vertices, no connected $K_2$ component, and at most one isolated vertex, $\operatorname{Dist^{\prime}}(\mu(G)) \le \operatorname{Dist^{\prime}}(G)$, exceeding their conjecture. We also prove analogous results about generalized Mycielskian graphs. Together with the work in 2022 of Boutin, Cockburn, Keough, Loeb, Perry, and Rombach this completes the conjecture of Alikhani and Soltani.

Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $\Delta$, then the choosability (or list chromatic number) of $G$ satisfies $\chi_{\ell}(G) = O \left ( \log \Delta \right )$. Currently, the best known upper bound for $\chi_{\ell}(G)$ is $(1 + o(1)) \frac{\Delta}{\log \Delta}$, which also holds for the much larger class of triangle-free graphs. We prove that for $\varepsilon = 10^{-3}$, every bipartite graph $G$ of sufficiently large maximum degree $\Delta$ satisfies $\chi_{\ell}(G) < (\frac{4}{5} -\varepsilon) \frac{\Delta}{\log \Delta}$. This improved upper bound suggests that list coloring is fundamentally different for bipartite graphs than for triangle-free graphs and hence gives a step toward solving the conjecture of Alon and Krivelevich., Comment: 8 pages

Let $G$ be a graph of maximum degree $\Delta$ which does not contain isolated vertices. An edge coloring $c$ of $G$ is called conflict-free if each edge's closed neighborhood includes a uniquely colored element. The least number of colors admitting such $c$ is called the conflict-free chromatic index of $G$ and denoted $\chi'_{\rm CF}(G)$. It is known that in general $\chi'_{\rm CF}(G)\leq 3 \lceil \log_2\Delta \rceil+1$, while there is a family of graphs, e.g. the complete graphs, for which $\chi'_{\rm CF}(G)\geq (1-o(1))\log_2\Delta$. In the present paper we provide the asymptotically tight upper bound $\chi'_{\rm CF}(G)\leq (1+o(1))\log_2\Delta$ for regular and nearly regular graphs, which in particular implies that the same bound holds a.a.s. for a random graph $G=G(n,p)$ whenever $p\gg n^{-\varepsilon}$ for any fixed constant $\varepsilon\in (0,1)$. Our proof is probabilistic and exploits classic results of Hall and Berge. This was inspired by our approach utilized in the particular case of complete graphs, for which we give a more specific upper bound. We also observe that almost the same bounds hold in the open neighborhood regime., Comment: 14 pages

A random set $S$ is $p$-spread if, for all sets $T$, $$\mathbb{P}(S \supseteq T) \leq p^{|T|}.$$ There is a constant $C>1$ large enough that for every graph $G$ with maximum degree $D$, there is a $C/D$-spread distribution on $(D+1)$-colorings of $G$. Making use of a connection between thresholds and spread distributions due to Frankston, Kahn, Narayanan, and Park, a palette sparsification theorem of Assadi, Chen, and Khanna follows., Comment: 18 pages

We consider the flexible list coloring problem, in which we have a graph $G$, a color list assignment $L:V(G) \rightarrow 2^{\mathbb N}$, and a set $U \subseteq V(G)$ of vertices such that each $u \in U$ has a preferred color $p(u) \in L(u)$. Given a constant $\varepsilon > 0$, the problem asks for an $L$-coloring of $G$ in which at least $\varepsilon |U|$ vertices in $U$ receive their preferred color. We use a method of reducible subgraphs to approach this problem. We develop a vertex-partitioning tool that, when used with a new reducible subgraph framework, allows us to define large reducible subgraphs. Using this new tool, we show that if $G$ has maximum average degree less than $\frac{11}{3}$, a list $L(v)$ of size $4$ at each $v \in V(G)$, and a set $U \subseteq V(G)$ of vertices with preferred colors, then there exists an $L$-coloring of $G$ for which at least $2^{-145} |U|$ vertices of $U$ receive their preferred color., Comment: 34 pages + appendix

A Latin tableau of shape $\lambda$ and type $\mu$ is a Young diagram of shape $\lambda$ in which each box contains a single positive integer, with no repeated integers in any row or column, and the $i$th most common integer appearing $\mu_i$ times. Over twenty years ago, Chow et al., in their study of a generalization of Rota's basis conjecture that they called the wide partition conjecture, conjectured a necessary and sufficient condition for the existence of a Latin tableau of shape $\lambda$ and type $\mu$. We report some computational evidence for this conjecture, and prove that the conjecture correctly characterizes, for any given $\lambda$, at least the first four parts of $\mu$., Comment: 20 pages

A colouring of a graph $G$ has clustering $k$ if the maximum number of vertices in a monochromatic component equals $k$. Motivated by recent results showing that many natural graph classes are subgraphs of the strong product of a graph with bounded treewidth and a path, this paper studies clustered colouring of strong products of two bounded treewidth graphs, where none, one, or both graphs have bounded degree. For example, in the case of two colours, if $n$ is the number of vertices in the product, then we show that clustering $\Theta(n^{2/3})$ is best possible, even if one of the graphs is a path. However, if both graphs have bounded degree, then clustering $\Theta(n^{1/2})$ is best possible. With three colours, if one of the graphs has bounded degree, then we show that clustering $\Theta(n^{3/7})$ is best possible. However, if neither graph has bounded degree, then clustering $\Omega(n^{1/2})$ is necessary. More general bounds for any given number of colours are also presented.

Let $R$ be a ring. The unitary addition Cayley graph of $R$, denoted $\mathcal{U}(R)$, is the graph with vertex $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ is a unit. We determine a formula for the clique number and chromatic number of such graphs when $R$ is a finite commutative ring. This includes the special case when $R$ is $\mathbb{Z}_n$, the integers modulo $n$, where these parameters had been found under the assumption that $n$ is even, or $n$ is a power of an odd prime. Additionally, we study the achromatic number of $\mathcal{U}( \mathbb{Z}_n )$ in the case that $n$ is the product of two primes. We prove that the achromatic number of $\mathcal{U} ( \mathbb{Z}_{3q})$ is equal to $\frac{3q+1}{2}$ when $q > 3$ is a prime. We also prove a lower bound that applies when $n = pq$ where $p$ and $q$ are distinct odd primes.

Let $G$ be a bridgeless cubic graph. The \textit{resistance} of $G$, denoted $r(G)$, is the minimum number of edges which can be removed from $G$ in order to render 3-edge-colourability. The \textit{oddness} of $G$, denoted $\omega(G)$, is the minimum number of odd components in a 2-factor of $G$. The \textit{colouring defect} of $G$ (or simply, the \textit{defect} of $G$), denoted $\mu_3(G)$, is the minimum number of edges not contained in any set of three perfect matchings of $G$. These three parameters are regarded as measurements of uncolourability of snarks, partly because any one of these parameters equal zero if and only if $G$ is 3-edge-colourable. It is also known that $r(G) \geq \omega(G)$ and that $\mu_3(G) \geq \frac{3}{2}\omega(G)$ \cite{fiol,jinsteffen}. We have shown that the ratio of oddness to resistance can be arbitrarily large for non-trivial snarks \cite{allie1}. It has also been shown that the ratio of the defect to oddness can be arbitrarily large for non-trivial snarks, although this result was only shown for graphs with oddness equal to 2 \cite{karabasetal}. In the same paper, the question was posed whether there exists non-trivial snarks for given resistance $r$ or given oddness $\omega$, and arbitrarily large defect. In this paper, we prove a stronger result: For any positive integers $r \geq 2$, even $\omega \geq r$, and $d \geq \frac{3}{2}\omega$, there exists a non-trivial snark $G$ with $r(G)=r$, $\omega(G)=\omega$ and $\mu_3(G) \geq d$.

It is shown that the following holds for each $\varepsilon >0$. For $G$ an $n$-vertex graph of maximum degree $D$, lists $S_v$ ($v\in V(G)$), and $L_v$ chosen uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $S_v$ (independent of other choices), \[ \mbox{$G$ admits a proper coloring $\sigma$ with $\sigma_v\in L_v$ $\forall v$} \] with probability tending to 1 as $D\to \infty$. When each $S_v $ is $\{1\dots D+1\}$, this is an asymptotically optimal version of the ``palette sparsification'' theorem of Assadi, Chen and Khanna that was proved in an earlier paper by the present authors., Comment: 37 pages, 0 figures. arXiv admin note: text overlap with arXiv:2306.00171

If a graph is $n$-colourable, then it obviously is $n'$-colourable for any $n'\ge n$. But the situation is not so clear when we consider multi-colourings of graphs. A graph is $(n,k)$-colourable if we can assign each vertex a $k$-subset of $\{1,2,\ldots,n\}$ so that adjacent vertices receive disjoint subsets. In this note we consider the following problem: if a graph is $(n,k)$-colourable, then for what pairs $(n', k')$ is it also $(n',k')$-colourable? This question can be translated into a question regarding multi-colourings of Kneser graphs, for which Stahl formulated a conjecture in 1976. We present new results, strengthen existing results, and in particular present much simpler proofs of several known cases of the conjecture., Comment: 17 pages

Let $H$ be a 2-regular graph and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. The "cycles plus $K_4$'s" problem is to show that $G$ is 4-colourable; this is a special case of the \emph{Strong Colouring Conjecture}. In this paper we reduce the "cycles plus $K_4$'s" problem to a specific 3-colourability problem. In the 3-colourability problem, vertex-disjoint triangles are glued (in a limited way) onto a disjoint union of triangles and paths of length at most 12, and we ask for 3-colourability of the resulting graph.

A 2-distance $k$-coloring of a graph $G$ is a proper $k$-coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of $G$ is the minimum $k$ such that $G$ has a 2-distance $k$-coloring, denoted by $\chi_2(G)$. In this paper, we show that $\chi_2(G) \leq 20$ for every planar graph $G$ with maximum degree at most six, which improves a former bound $\chi_2(G) \leq 21$., Comment: 12 pages

The Erd\H{o}s-Lov\'asz Tihany Conjecture states that any $G$ with chromatic number $\chi(G) = s + t - 1 > \omega(G)$, with $s,t \geq 2$ can be split into two vertex-disjoint subgraphs of chromatic number $s, t$ respectively. We prove this conjecture for pairs $(s, t)$ if $t \leq s + 2$, whenever $G$ has a $K_s$, and for pairs $(s, t)$ if $t \leq 4 s - 3$, whenever $G$ contains a $K_s$ and is claw-free. We also prove the Erd\H{o}s Lov\'asz Tihany Conjecture for the pair $(3, 10)$ for claw-free graphs., Comment: 18 pages, 3 figures

The $c$-strong chromatic number of a hypergraph is the smallest number of colours needed to colour its vertices so that every edge sees at least $c$ colours or is rainbow. We show that every $t$-intersecting hypergraph has bounded $(t + 1)$-strong chromatic number, resolving a problem of Blais, Weinstein and Yoshida. In fact, we characterise when a $t$-intersecting hypergraph has large $c$-strong chromatic number for $c\geq t+2$. Our characterisation also applies to hypergraphs which exclude sunflowers with specified parameters., Comment: 13 pages

A well known problem from an excellent book of Lov\'asz states that any hypergraph with the property that no pair of hyperedges intersect in exactly one vertex can be properly 2-colored. Motivated by this as well as recent works of Keszegh and of Gy\'arf\'as et al we study the $1$-intersection graph of a hypergraph. The $1$-intersection graph encodes those pairs of hyperedges in a hypergraph that intersect in exactly one vertex. We prove for $k\in\{2,4\}$ that all hypergraphs whose $1$-intersection graph is $k$-partite can be properly $k$-colored.

A graph/multigraph $G$ is locally irregular if endvertices of every its edge possess different degrees. The locally irregular edge coloring of $G$ is its edge coloring with the property that every color induces a locally irregular sub(multi)graph of $G$; if such a coloring of $G$ exists, the minimum number of colors to color $G$ in this way is the locally irregular chromatic index of $G$ (denoted by ${\rm lir}(G)$). We state the following new problem: given a connected graph $G$ distinct from $K_2$ or $K_3$, what is the minimum number of edges of $G$ to be doubled such that the resulting multigraph is locally irregular edge colorable (with no monochromatic multiedges) using at most two colors? This problem is closely related to several open conjectures (like the Local Irregularity Conjecture for graphs and 2-multigraphs, or (2, 2)-Conjecture) and other similar edge coloring concepts. We present the solution of this problem for several graph classes: paths, cycles, trees, complete graphs, complete $k$-partite graphs, split graphs and powers of cycles. Our solution for complete $k$-partite graphs ($k>1$) and powers of cycles (which are not complete graphs) shows that, in this case, the locally irregular chromatic index equals 2. We also consider this problem for special families of cacti and prove that the minimum number of edges in a graph whose doubling yields an local irregularly colorable multigraph does not have a constant upper bound not only for locally irregular uncolorable cacti., Comment: 37 pages, 8 figures

The 2-distance coloring of a graph $G$ is equivalent to the proper coloring of its square graph $G^2$, it is a special distance labeling problem. DP-coloring (or "Correspondence coloring") was introduced by Dvo\v{r}\'ak and Postle in 2018, to answer a conjecture of list coloring proposed by Borodin. In recent years, many researches pay attention to the DP-coloring of planar graphs with some restriction in cycles. We study the DP-coloring of the square of subcubic graphs in terms of maximum average degree $\rm{mad}(G)$, and by the discharging method, we showed that: for a subcubic graph $G$, if $\rm{mad}(G)<9/4$, then $G^2$ is DP-5-colorable; if $\rm{mad}(G)<12/5$, then $G^2$ is DP-6-colorable. And the bound in the first result is sharp., Comment: 7 pages, 3 figures

We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta(G)+2\left\lceil \frac{|V(G)|}{\Delta(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result says that if $\Delta(G)\ge \frac{1}{2}|V(G)|$, then $G$ has a total coloring using at most $\Delta(G)+4$ colors. When $G$ is regular and has a sufficient number of vertices, we can actually save an additional two colors. Specifically, we prove that for any $0<\varepsilon <1$, there exists $n_0\in \mathbb{N}$ such that: if $G$ is an $r$-regular graph on $n \ge n_0$ vertices with $r\ge \frac{1}{2}(1+\varepsilon) n$, then $\chi_T(G) \le \Delta(G)+2$. This confirms the Total Coloring Conjecture for such graphs $G$.

The reconfiguration graph of the $k$-colorings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_{k}(G)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R_{\ell}(G)$ is connected for all $\ell \geq \chi(G)$+1. We demonstrate how to use the modular decomposition of a graph class to prove that the graphs in the class are recolorable. In particular, we prove that every ($P_5$, diamond)-free graph, every ($P_5$, house, bull)-free graph, and every ($P_5$, $C_5$, co-fork)-free graph is recolorable. A graph is prime if it cannot be decomposed by modular decomposition except into single vertices. For a prime graph $H$, we study the complexity of deciding if $H$ is $k$-colorable and the complexity of deciding if there exists a path between two given $k$-colorings in $R_{k}(H)$. Suppose $\mathcal{G}$ is a hereditary class of graphs. We prove that if every blowup of every prime graph in $\mathcal{G}$ is recolorable, then every graph in $\mathcal{G}$ is recolorable., Comment: 13 pages

A total coloring of a graph colors all its elements, vertices and edges, with no adjacency conflicts. The Total Coloring Conjecture (TCC) is a sixty year old challenge, says that every graph admits a total coloring with at most maximum degree plus two colors, and many graph parameters have been studied in connection with its validity. If a graph admits a total coloring with maximum degree plus one colors, then it is Type 1, whereas it is Type 2, in case it does not admit a total coloring with maximum degree plus one colors but it does satisfy the TCC. Cavicchioli, Murgolo and Ruini proposed in 2003 the hunting for a Type 2 snark with girth at least 5. Brinkmann, Preissmann and Sasaki in 2015 conjectured that there is no Type 2 cubic graph with girth at least 5. We investigate the total coloring of fullerene nanodiscs, a class of cubic planar graphs with girth 5 arising in Chemistry. We prove that the central layer of an arbitrary fullerene nanodisc is 4-total colorable, a necessary condition for the nanodisc to be Type 1. We extend the obtained 4-total coloring to a 4-total coloring of the whole nanodisc, when the radius satisfies r = 5 + 3k, providing an infinite family of Type 1 nanodiscs., Comment: 25 pages, 21 Figures

The paintability of a graph is a coloring parameter defined in terms of an online list coloring game. In this paper we ask, what is the paintability of a graph $G$ of maximum degree $\Delta$ and chromatic number $r$? By considering the Alon-Tarsi number of $G$, we prove that the paintability of $G$ is at most $\left(1 - \frac{1}{4r+1} \right ) \Delta + 2$. We also consider the DP-paintability of $G$, which is defined in terms of an online DP-coloring game. By considering the strict type $3$ degeneracy parameter recently introduced by Zhou, Zhu, and Zhu, we show that when $r$ is fixed and $\Delta$ is sufficiently large, the DP-paintability of $G$ is at most $\Delta - \Omega( \sqrt{\Delta \log \Delta})$., Comment: 13 pages, a terminology error was corrected

We show that the chromatic index of a hypergraph $\mathcal{H}$ satisfies Berge-F\"uredi conjectured bound $\mathrm{q}(\mathcal{H})\le \Delta([\mathcal{H}]_2)+1$ under certain hypotheses on the antirank $\mathrm{ar}(\mathcal{H})$ or on the maximum degree $\Delta(\mathcal{H})$. This provides sharp information in connection with Erd\H{o}s-Faber-Lov\'asz Conjecture which deals with the coloring of a family of cliques that intersect pairwise in at most one vertex.

This paper is concerned with two conjectures which are intimately related. The first is a generalization to hypergraphs of Vizing's Theorem on the chromatic index of a graph and the second is the well-known conjecture of Erd\H{o}s, Faber and Lov\'asz which deals with the problem of coloring a family of cliques intersecting in at most one vertex. We are led to study a special class of uniform and linear hypergraphs for which a number of properties are established.

We say that a graph $G$ is chromatic-choosable when its list chromatic number $\chi_{\ell}(G)$ is equal to its chromatic number $\chi(G)$. Chromatic-choosability is a well-studied topic, and in fact, some of the most famous results and conjectures related to list coloring involve chromatic-choosability. In 2002 Ohba showed that for any graph $G$ there is an $N \in \mathbb{N}$ such that the join of $G$ and a complete graph on at least $N$ vertices is chromatic-choosable. The Ohba number of $G$ is the smallest such $N$. In 2014, Noel suggested studying the Ohba number, $\tau_{0}(a,b)$, of complete bipartite graphs with partite sets of size $a$ and $b$. In this paper we improve a 2009 result of Allagan by showing that $\tau_{0}(2,b) = \lfloor \sqrt{b} \rfloor - 1$ for all $b \geq 2$, and we show that for $a \geq 2$, $\tau_{0}(a,b) = \Omega( \sqrt{b} )$ as $b \rightarrow \infty$. We also initiate the study of some relaxed versions of the Ohba number of a graph which we call generalized Ohba numbers. We present some upper and lower bounds of generalized Ohba numbers of complete bipartite graphs while also posing some questions., Comment: 15 pages

We say a graph $H$ is $r$-rainbow-uncommon if the maximum number of rainbow copies of $H$ under an $r$-coloring of $E(K_n)$ is asymptotically (as $n \to \infty$) greater than what is expected from uniformly random $r$-colorings. Via explicit constructions, we show that for $H\in\{K_3,K_4, K_5\}$, $H$ is $r$-rainbow-uncommon for all $r\geq {|V(H)|\choose 2}$. We also construct colorings to show that for $t \geq 6$, $K_t$ is $r$-rainbow-uncommon for sufficiently large $r$., Comment: 9 pages, 2 figures

A vertex colouring of some graph is called perfect if each vertex of colour $i$ has the same number $a_{ij}$ of neighbours of colour $j$. Here we determine all perfect colourings of the edge graphs of the hypercube in dimensions 4 and 5 by two and three colours, respectively. For comparison we list all perfect colourings of the edge graphs of the simplex in dimensions 4 and 5, respectively., Comment: 10 pages, 7 figures

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. It was conjectured that every connected graph is edge decomposable to $3$ locally irregular subgraphs, unless it belongs to a certain family of exceptions, including graphs of small maximum degrees, which are not decomposable to any number of such subgraphs. Recently Sedlar and \v{S}krekovski exhibited a counterexample to the conjecture, which necessitates a decomposition to (at least) $4$ locally irregular subgraphs. We prove that every $d$-regular graph with $d$ large enough, i.e. $d\geq 54000$, is decomposable to $4$ locally irregular subgraphs. Our proof relies on a mixture of a numerically optimized application of the probabilistic method and certain deterministic results on degree constrained subgraphs due to Addario-Berry, Dalal, McDiarmid, Reed, and Thomason, and to Alon and Wei, introduced in the context of related problems concerning irregular subgraphs., Comment: 16 pages

A proper abelian coloring of a cubic graph G by a finite abelian group A is any proper edge-coloring of G by the non-zero elements of A such that the sum of the colors of the three edges incident to any vertex v of G equals zero. It is known that cyclic groups of order smaller than 10 do not color all bridgeless cubic graphs, and that all abelian groups of order at least 12 do. This leaves the question open for the four so called exceptional groups Z4 x Z2, Z3 x Z3, Z10 and Z11 for snarks. It is conjectured in literature that every cubic graph has a proper abelian coloring by each exceptional group and it is further known that the existence of a proper Z4 x Z2-coloring of G implies the existence of a proper coloring of G by all the remaining exceptional groups. In this paper, we give a characterization of a proper Z4 x Z2-coloring in terms of the existence of a matching M in a 2-factor F of G with particular properties. Moreover, in order to modify an arbitrary matching M so that it meets the requirements of the characterization, we first introduce an incidence structure of the cycles of F in relation to the cycles of G - M. Further, we provide a sufficient condition under which M can be modified into a desired matching in terms of particular properties of the introduced incidence structure. We conclude the paper by applying the results to some oddness two snarks, in particular to permutation snarks. We believe that the approach of this paper with some additional refinements extends to larger classes of snarks, if not to all in general., Comment: 26 pages, 13 figures

An odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, there is a color appearing an odd number of times in $N_G(v)$. Odd coloring of graphs was studied intensively in recent few years. In this paper, we introduce the notion of a strong odd coloring, as not only a strengthened version of odd coloring, but also a relaxation of square coloring. A strong odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, if a color appears in $N_G(v)$, then it appears an odd number of times in $N_G(v)$. We denote by $\chi_{so}(G)$ the smallest integer $k$ such that $G$ admits a strong odd coloring with $k$ colors. We prove that if $G$ is a graph with $mad(G)\le\frac{20}{7}$, then $\chi_{so}(G)\le \Delta(G)+4$, and the bound is tight. We also prove that if $G$ is a graph with $mad(G)\le\frac{30}{11}$ and $\Delta(G)\ge 4$, then $\chi_{so}(G)\le \Delta(G)+3$., Comment: 21 pages, 12 figures

For a graph $G$ and a list assignment $L$ with $|L(v)|=k$ for all $v$, an $L$-packing consists of $L$-colorings $\varphi_1,\cdots,\varphi_k$ such that $\varphi_i(v)\ne\varphi_j(v)$ for all $v$ and all distinct $i,j\in\{1,\ldots,k\}$. Let $\chi^{\star}_{\ell}(G)$ denote the smallest $k$ such that $G$ has an $L$-packing for every $L$ with $|L(v)|=k$ for all $v$. Let $\mathcal{P}_k$ denote the set of all planar graphs with girth at least $k$. We show that (i) $\chi^{\star}_{\ell}(G)\le 8$ for all $G\in \mathcal{P}_3$ and (ii) $\chi^{\star}_{\ell}(G)\le 5$ for all $G\in \mathcal{P}_4$ and (iii) $\chi^{\star}_{\ell}(G)\le 4$ for all $G\in \mathcal{P}_5$. Part (i) makes progress on a problem of Cambie, Cames van Batenburg, Davies, and Kang. We also construct outerplanar graphs $G$ such that $\chi^{\star}_{\ell}(G)=4$, which matches the known upper bound $\chi^{\star}_{\ell}(G)\le 4$ for all outerplanar graphs. Finally, we consider the analogue of $\chi^{\star}_{\ell}$ for correspondence coloring, $\chi^{\star}_c$. In fact, all bounds stated above for $\chi^{\star}_{\ell}$ also hold for $\chi^{\star}_c$., Comment: 20 pages, 15 figures, 2nd version incorporates reviewer feedback, to appear in J. of Graph Theory; 3rd version updates title to match the final journal version

An improper interval (edge) coloring of a graph $G$ is an assignment of colors to the edges of $G$ satisfying the condition that, for every vertex $v \in V(G)$, the set of colors assigned to the edges incident with $v$ forms an integral interval. An interval coloring is $k$-improper if at most $k$ edges with the same color all share a common endpoint. The minimum integer $k$ such that there exists a $k$-improper interval coloring of the graph $G$ is the interval coloring impropriety of $G$, denoted by $\mu_{int}(G)$. In this paper, we provide a construction of an interval coloring of a subclass of complete multipartite graphs. This provides additional evidence to the conjecture by Casselgren and Petrosyan that $\mu_{int}(G)\leq 2$ for all complete multipartite graphs $G$. Additionally, we determine improved upper bounds on the interval coloring impropriety of several classes of graphs, namely 2-trees, iterated triangulations, and outerplanar graphs. Finally, we investigate the interval coloring impropriety of the corona product of two graphs, $G\odot H$., Comment: 17 pages, 8 figures, 7 tables

A normal 5-edge-coloring of a cubic graph is a coloring such that for every edge the number of distinct colors incident to its end-vertices is 3 or 5 (and not 4). The well known Petersen Coloring Conjecture is equivalent to the statement that every bridgeless cubic graph has a normal 5-edge-coloring. All 3-edge-colorings of a cubic graph are obviously normal, so in order to establish the conjecture it is sufficient to consider only snarks. In our previous paper [J. Sedlar, R. \v{S}krekovski, Normal 5-edge-coloring of some snarks superpositioned by the Petersen graph, Applied Mathematics and Computation 467 (2024) 128493], we considered superpositions of any snark G along a cycle C by two simple supervertices and by the superedge obtained from the Petersen graph, but only for some of the possible ways of connecting supervertices and superedges. The present paper is a continuation of that paper, herein we consider superpositions by the Petersen graph for all the remaining connections and establish that for all of them the Petersen Coloring Conjecture holds., Comment: 19 pages, 10 figures

In 2020, Behr defined the problem of edge coloring of signed graphs and showed that every signed graph $(G, \sigma)$ can be colored using exactly $\Delta(G)$ or $\Delta(G) + 1$ colors, where $\Delta(G)$ is the maximum degree in graph $G$. In this paper, we focus on products of signed graphs. We recall the definitions of the Cartesian, tensor, strong, and corona products of signed graphs and prove results for them. In particular, we show that $(1)$ the Cartesian product of $\Delta$-edge-colorable signed graphs is $\Delta$-edge-colorable, $(2)$ the tensor product of a $\Delta$-edge-colorable signed graph and a signed tree requires only $\Delta$ colors and $(3)$ the corona product of almost any two signed graphs is $\Delta$-edge-colorable. We also prove some results related to the coloring of products of signed paths and cycles.

The reconfiguration graph of the $k$-colorings, denoted $R_k(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_k(G)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R_{\ell}(G)$ is connected for all $\ell\geq \chi(G)$+1. In this paper, we study the recolorability of several graph classes restricted by forbidden induced subgraphs. We prove some properties of a vertex-minimal graph $G$ which is not recolorable. We show that every (triangle, $H$)-free graph is recolorable if and only if every (paw, $H$)-free graph is recolorable. Every graph in the class of $(2K_2,\ H)$-free graphs, where $H$ is a 4-vertex graph except $P_4$ or $P_3$+$P_1$, is recolorable if $H$ is either a triangle, paw, claw, or diamond. Furthermore, we prove that every ($P_5$, $C_5$, house, co-banner)-free graph is recolorable., Comment: 17 pages

For every $r\ge13$, we show every 1-planar graph $G$ with $\Delta(G)\le r$ has an equitable $r$-coloring., Comment: 9 figures

Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent vertices u and v have different colors c(u) and c(v). A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in its closed neighborhood. The maximum number of colors k admitting b-coloring of G is the b-chromatic number. We present two separate approaches to the conjecture posed by Blidia et. al that the b-chromatic number equals to d+1 for every d-regular graph of girth at least five except the Petersen graph., Comment: 16 pages, 2 figures, 13 references

A graph is $(\mathcal{I}, \mathcal{F})$-colorable if its vertex set can be partitioned into two subsets, one of which is an independent set, and the other induces a forest. In this paper, we prove that every planar graph without $5^{-}$-cycles at distance less than $3$ is $(\mathcal{I}, \mathcal{F})$-colorable., Comment: 11 pages, 3 figures