Abstract: Based on the algorithmic proof of Lovasz local lemma due to Moser and Tardos, the works of Grytczuk et al on words, and Dujmovic et al on colorings, Esperet and Parreau developed a framework to prove upper bounds for several chromatic numbers (in particular acyclic chromatic index, star chromatic number and Thue chromatic number) using the so-called \emph{entropy compression method} Inspired by this work, we propose a more general framework and a better analysis This leads to improved upper bounds on chromatic numbers and indices In particular, every graph with maximum degree $\Delta$ has an acyclic chromatic number at most $\frac{3}{2}\Delta^{\frac43} + O(\Delta)$ Also every planar graph with maximum degree $\Delta$ has a facial Thue choice number at most $\Delta + O(\Delta^\frac 12)$ and facial Thue choice index at most $10$

Abstract: Based on a non-rigorous formalism called the "cavity method", physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random $k$-SAT or random graph $k$-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called "condensation" [Krzakala et al., PNAS 2007]. The existence of this phase transition appears to be intimately related to the difficulty of proving precise results on, e.g., the $k$-colorability threshold as well as to the performance of message passing algorithms. In random graph $k$-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem. In this paper we prove this conjecture for $k$ exceeding a certain constant $k_0$.

Abstract: One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdi¾?s-Renyi random graph Gn,p is around p~logn+loglognn. Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3-uniform hypergraph by connecting 3-uniform hypergraphs to edge-colored graphs.

Abstract: The algorithm for Lov\'{a}sz Local Lemma by Moser and Tardos gives a constructive way to prove the existence of combinatorial objects that satisfy a system of constraints. We present an alternative probabilistic analysis of the algorithm that does not involve reconstructing the history of the algorithm. We apply our technique to improve the best known upper bound to acyclic chromatic index. Specifically we show that a graph with maximum degree $\Delta$ has an acyclic proper edge coloring with at most $\lceil 3.74(\Delta-1)\rceil+1 $ colors, whereas the previously known best bound was $4(\Delta-1)$. The same technique is also applied to improve corresponding bounds for graphs with bounded girth. An interesting aspect of the latter application is that the probability of the "undesirable" events do not have a uniform upper bound, i.e., it constitutes a case of the asymmetric Lov\'{a}sz Local Lemma.

Abstract: We show that an effective version of Siegel's Theorem on finiteness of integer solutions for a specific algebraic curve and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the k >= 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. The hardness, and thus the dichotomy, holds even when restricted to planar graphs. A special case of this result is that counting edge k-colorings is #P-hard over planar 3-regular multigraphs for all k >= 3. In fact, we prove that counting edge k-colorings is #P-hard over planar r-regular multigraphs for all k >= r >= 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for Holant(f) depends on the fact that a specific polynomial p(x, y) has an explicitly listed finite set of integer solutions, and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials.

Abstract: A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge coloring of a path is nonrepetitive if the sequence of colors of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colorings for arbitrarily long paths using only three colors. A recent generalization of this concept implies that we may obtain such colorings even if we are forced to choose edge colors from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these basic ideas, Havet, Jendrol', Sotak, and Skrabul'akova proved that for each plane graph, eight colors are sufficient to provide an edge coloring so that every facial path is nonrepetitively colored. In this article, we prove that the same is possible from lists, provided that these have size at least 12. We thus improve the previous bound of 291 (proved by means of the Lovasz Local Lemma). Our approach is based on the Moser–Tardos entropy-compression method and its recent extensions by Grytczuk, Kozik, and Micek, and by Dujmovic, Joret, Kozik, and Wood.

Abstract: Let Δ ≥ 4 be an integer. In this note, we prove that every planar graph with maximum degree Δ and girth at least 1 Δ+46 is strong (2Δ−1)-edgecolorable, that is best possible (in terms of number of colors) as soon as G contains two adjacent vertices of degree Δ. This improves [6] when Δ ≥ 6.

Abstract: An edge-coloring of a graph G with colors 1,...,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In 1991, Erdi¾?s constructed a bipartite graph with 27 vertices and maximum degree 13 that has no interval coloring. Erdi¾?s's counterexample is the smallest in a sense of maximum degree known bipartite graph that is not interval colorable. On the other hand, in 1992, Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this article, we give some methods for constructing of interval non-edge-colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs that have no interval coloring, contain 20, 19, 21 vertices and have maximum degree 11, 12, 13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.

Abstract: We give a probabilistic analysis of a Moser-type algorithm for the Lovasz Local Lemma (LLL), adjusted to search for acyclic edge colorings of a graph. We thus improve the best known upper bound to acyclic chromatic index, also obtained by analyzing a similar algorithm, but through the entropic method (basically counting argument). Specifically we show that a graph with maximum degree $\Delta$ has an acyclic proper edge coloring with at most $\lceil 3.74(\Delta-1)\rceil+1 $ colors, whereas, previously, the best bound was $4(\Delta-1)$. The main contribution of this work is that it comprises a probabilistic analysis of a Moser-type algorithm applied to events pertaining to dependent variables.

Abstract: In online list coloring (introduced by Zhu and by Schauz in 2009), on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset to receive that color. The paint number of a graph G is the least k such that there is an algorithm to produce a successful coloring with no vertex being shown more than k times; it is at least the choice number. We study paintability of joins with complete or empty graphs, obtaining a partial result toward the paint analogue of Ohba’s Conjecture. We also determine upper and lower bounds on the paint number of complete bipartite graphs and characterize 3-paintcritical graphs.

Abstract: Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the interior along surfaces we can adapt the degrees along the boundary of that surface. More efficient is a self-cobordism of G with itself with a host graph discretizing the product of G with an interval. It follows from the fact that Euler curvature is zero everywhere for three dimensional geometric graphs, that the odd degree edge set O is a cycle and so a boundary if H is simply connected. A reduction to minimal colouring would imply the four colour theorem. The method is expected to give a reason "why 4 colours suffice" and suggests that every two dimensional geometric graph of arbitrary degree and orientation can be coloured by 5 colours: since the projective plane can not be a boundary of a 3-dimensional graph and because for higher genus surfaces, the interior H is not simply connected, we need in general to embed a surface into a 4-dimensional simply connected graph in order to colour it. This explains the appearance of the chromatic number 5 for higher degree or non-orientable situations, a number we believe to be the upper limit. For every surface type, we construct examples with chromatic number 3,4 or 5, where the construction of surfaces with chromatic number 5 is based on a method of Fisk. We have implemented and illustrated all the topological aspects described in this paper on a computer. So far we still need human guidance or simulated annealing to do the refinements in the higher dimensional host graph.

Abstract: The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this article, we prove that deciding whether this number is at most four for a given cubic bridgeless graph is NP-complete. We also construct an infinite family F of snarks cyclically 4-edge-connected cubic graphs of girth at least 5 and chromatic index 4 whose edge-set cannot be covered by four perfect matchings. Only two such graphs were known. It turns out that the family F also has interesting properties with respect to the shortest cycle cover problem. The shortest cycle cover of any cubic bridgeless graph with m edges has length at least , and we show that this inequality is strict for graphs of F. We also construct the first known snark with no cycle cover of length less than .

Abstract: In this paper, we give a polynomial time algorithm which determines if a given triangle-free graph with no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists.

Abstract: Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is familiar to us by the name of "total coloring". Total coloring is a coloring of $$V\cup {E}$$ V ? E such that no two adjacent or incident elements receive the same color. In other words, total chromatic number of $$G$$ G is the minimum number of disjoint vertex independent sets covering a total graph of $$G$$ G . Here, let $$G$$ G be a planar graph with $$\varDelta \ge 8$$ Δ ? 8 . We proved that if for every vertex $$v\in V$$ v ? V , there exists two integers $$i_{v},j_{v} \in \{3,4,5,6,7,8\}$$ i v , j v ? { 3 , 4 , 5 , 6 , 7 , 8 } such that $$v$$ v is not incident with intersecting $$i_v$$ i v -cycles and $$j_v$$ j v -cycles, then the vertex chromatic number of total graph of $$G$$ G is $$\varDelta +1$$ Δ + 1 , i.e., the total chromatic number of $$G$$ G is $$\varDelta +1$$ Δ + 1 .

Abstract: Let G and H be two graphs. A proper vertex coloring of G is called a dynamic coloring, if for every vertex v with degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic coloring with k colors denoted by χ2(G). We denote the cartesian product of G and H by G¤H. In this paper, we prove that if G and H are two graphs and δ(G) ≥ 2, then χ2(G¤H) ≤ max(χ2(G), χ(H)). We show that for every two natural numbers m and n, m,n ≥ 2, χ2(Pm¤Pn) = 4. Also, among other results it is shown that if 3|mn, then χ2(Cm¤Cn) = 3 and otherwise χ2(Cm¤Cn) = 4.

Abstract: Let $G=(V,E)$ be a simple graph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\Delta +1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$ colors. Vizing's theorem gives a bound of $\frac{\Delta}{\Delta+1}|E|$. This is known to be tight for cliques $K_{\Delta+1}$ when $\Delta$ is even. However, for $\Delta=3$ it was improved to $\frac{26}{31}|E|$ by Albertson and Haas [Discrete Math., 148 (1996), pp. 1--7] and later to $\frac{6}7|E|$ by Rizzi [Discrete Math., 309 (2009), pp. 4166--4170]. It is tight for $B_3$, the graph isomorphic to a $K_4$ with one edge subdivided. We improve previously known bounds for $\Delta\in\{3,\ldots,7\}$, under the assumption that for $\Delta=3,4,6$, graph $G$ is not isomorphic to $B_3$, $K_5$, and $K_7$, respectively. For $\Delta \geq 4$ these are the first results which improve over the Vizing's bound. We also show a new bound for subcubic multigraphs not isomorphic to $K_3$ with one ...

Abstract: Fix positive integers p and q with 2 q p . An edge coloring of the complete graph Kn is said to be a (p,q)-coloring if every Kp receives at least q different colors. The function f (n,p,q) is the minimum number of colors that are needed for Kn to have a (p,q)-coloring. This function

Abstract: An edge-colored graph G, where adjacent edges may have the same color, is rainbow connected if every two vertices of G are connected by a path whose edges have distinct colors. A graph G is d-rainbow connected if one can use d colors to make G rainbow connected. For integers n and d let t(n, d) denote the minimum size (number of edges) in d-rainbow connected graphs of order n. Schiermeyer got some exact values and upper bounds for t(n, d). However, he did not present a lower bound of t(n, d) for $${3 \leq d < \lceil\frac{n}{2}\rceil}$$ . In this paper, we improve his lower bound of t(n, 2), and get a lower bound of t(n, d) for $${3 \leq d < \lceil\frac{n}{2}\rceil}$$ .