Improved upper bounds on acyclic edge colorings

Abstract

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors. The acyclic chromatic index of a graph G, denoted by a′(G), is the minimum number k such that there is an acyclic edge coloring using k colors. It is known that a′(G) ≤ 16Δ for every graph G where Δ denotes the maximum degree of G. We prove that a′(G) < 13.8Δ for an arbitrary graph G. We also reduce the upper bounds of a′(G) to 9.8Δ and 9Δ with girth 5 and 7, respectively.