A coloring of the vertices of a graph is called acyclic if the ends of each edge are colored in distinct colors, and there are no two-colored cycles. Suppose each face of rank k , k ≥ 4 , in a map on a surface S is replaced by the clique having the same number of vertices. It is proved in [1] that the resulting pseudograph admits an acyclic coloring with the number of colors depending linearly ...

Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, 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 entropy compression method. Inspired by this work, we propose a more general framework and a better analysis. This l...

In this report we obtain (good) upper bounds on the acyclic chromatic index of graphs obtained as the tensor product of two (or more) graphs, in terms of the acyclic chromatic indices of those factor graphs in the tensor product. Our results assume that optimal colourings for the factor graphs are given and our algorithms extend these colourings to obtain good (close to optimal) colourings of t...

The algorithm for Lovász 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 counting witness-trees. We then apply our approach to Acyclic Edge Coloring to obtain a direct probabilistic proof that a graph with maximum ...

Proof. We denote V -[p] = { 1 , . . . , p}, A(G) is the set of acyclic orientations of G and a(G) = IA(G)I is their number. An n-coloring of G, c: V---> [n] induces an acyclic orientation DceA(G) as follows: If [x,y]eE is an edge, where c(x) > c(y) then in Dc this edge is oriented from x to y. Every acyclic orientation D ~ A(G) defines a partial order on V, which we denote by i>o. If D e A(G), ...

‎For a coloring $c$ of a graph $G$‎, ‎the edge-difference coloring sum and edge-sum coloring sum with respect to the coloring $c$ are respectively‎ ‎$sum_c D(G)=sum |c(a)-c(b)|$ and $sum_s S(G)=sum (c(a)+c(b))$‎, ‎where the summations are taken over all edges $abin E(G)$‎. ‎The edge-difference chromatic sum‎, ‎denoted by $sum D(G)$‎, ‎and the edge-sum chromatic sum‎, ‎denoted by $sum S(G)$‎, ‎a...

Coloring graphs is one of important and frequently used topics in diverse sciences. In the majority of the articles, it is intended to find a proper bound for vertex coloring, edge coloring or total coloring in the graph. Although it is important to find a proper algorithm for graph coloring, it is hard and time-consuming too. In this paper, a new algorithm for vertex coloring, edge coloring an...

Acyclic-coloring of a graph G = (V; E) is a partitioning of V , such that the induced subgraph of each partition is acyclic. The minimum number of such partitions of V is deened as the vertex arboricity of G. An O(n) algorithm (n = jV j) for acyclic-coloring of planar graphs with 3 colors is presented. Next, an O(n 2) heuristic is proposed which produces a valid acyclic-2-coloring of a planar g...

Acyclic-coloring of a graph G = (V;E) is a partitioning of V , such that the induced subgraph of each partition is acyclic. The minimumnumber of such partitions of V is de ned as the vertex arboricity of G. An O(n) algorithm (n = jV j) for acyclic-coloring of planar graphs with 3 colors is presented. Next, an O(n) heuristic is proposed which produces a valid acyclic-2-coloring of a planar graph...