A strong edge-coloring of a graph is a function that assigns to each edge a color such that every two distinct edges that are adjacent or adjacent to a same edge receive different colors. The strong chromatic index χs(G) of a graph G is the minimum number of colors used in a strong edge-coloring of G. From a primal-dual point of view, there are three natural lower bounds of χs(G), that is σ(G) ...

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. In 1978, it was conjectured that ∆(G) + 2 colors suffice for an acyclic edge coloring of every graph G [6]. The conjecture has been verified for several classes of graphs, however, the best known upper bound for as special class as planar graphs are, is ∆+12 [2]. In this paper, we study simple planar graph...

An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of products of paths and cycles are considered. We determine the acyclic chromatic numbers of three such products: grid graphs...

A proper edge coloring of a graph G is said to be acyclic if every cycle of G receives at least three colors. The acyclic edge chromatic number of G, denoted a(G), is the least number of colors in an acyclic edge coloring of G. Alon, Sudakov and Zaks [Acyclic edge coloring of graphs, J. Graph Theory 37 (2001), 157-167] conjectured that a(G) ≤ ∆(G) + 2 holds for any graph G. In present paper, we...

An edge-coloring of a graph G1⁄4 ðV ; EÞ is a function c that assigns an integer c(e) (called color) in f 0;1;2;...g to every edge eAE so that adjacent edges are assigned different colors. An edge-coloring is compact if the colors of the edges incident to every vertex form a set of consecutive integers. The deficiency problem is to determine the minimum number of pendant edges that must be adde...

Throughout, by a graph we mean a simple undirected graph, where the degree of a vertex is its number of neighbors, and a d-coloring is a function assigning each vertex one of d colors so that adjacent vertices are mapped to different colors. This paper examines measurable analogues of Brooks’s Theorem. While a straightforward compactness argument extends Brooks’s Theorem to infinite graphs, suc...

An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding a...

A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some new classes of light graphs in 1-planar graphs with the bounded degree are found. Theref...