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). Fiamčik (1978) and independently Alon, Sudakov, and Zaks (2001) conjectured that a′(G)≤Δ(G)+2, where Δ(G) denotes the maximum degree of G. The best known general bound is a′(G)≤4(Δ(G)−1) due to Esperet and Parreau (2013). We apply a generalization of the Lovász Local Lemma to show that if G contains no copy of a given bipartite graph H, then a′(G)≤3Δ(G)+o(Δ(G)). Moreover, for every ε>0, there exists a constant c such that if g(G)≥c, then a′(G)≤(2+ε)Δ(G)+o(Δ(G)), where g(G) denotes the girth of G.