New bounds for the acyclic chromatic index

An edge coloring of a graph G is called an acyclic edge coloring if it is proper and every cycle in G contains edges of at least three different colors. The least number of colors needed for an acyclic edge coloring of G is called the acyclic chromatic index of G and is denoted by a'(G). Fiamcik (1978) and independently Alon, Sudakov, and Zaks (2001) conjectured that a'(G) <= Delta(G) + 2, where A(G) denotes the maximum degree of G. The best known general bound is a'(G) <= 4(Delta(G) - 1) due to Esperet and Parreau (2013). We apply a generalization of the Lovasz Local Lemma to show that if G contains no copy of a given bipartite graph H, then a'(G) <= 3 Delta 4(G) + o(Delta(G)). Moreover, for.every epsilon > 0, there exists a constant c such that if g(G) >= c, then a'(G) <= (2 + epsilon) Delta (G) + o(Delta(G)), where g (G) denotes the girth of G. (C) 2016 Elsevier B.V. All rights reserved.