New bounds for the acyclic chromatic index

An edge coloring of a graph GG is called an acyclic edge coloring   if it is proper and every cycle in GG contains edges of at least three different colors. The least number of colors needed for an acyclic edge coloring of GG is called the acyclic chromatic index   of GG and is denoted by a′(G)a′(G). Fiamčik (1978) and independently Alon, Sudakov, and Zaks (2001) conjectured that a′(G)≤Δ(G)+2a′(G)≤Δ(G)+2, where Δ(G)Δ(G) denotes the maximum degree of GG. The best known general bound is a′(G)≤4(Δ(G)−1)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 GG contains no copy of a given bipartite graph HH, then a′(G)≤3Δ(G)+o(Δ(G))a′(G)≤3Δ(G)+o(Δ(G)). Moreover, for every ε>0ε>0, there exists a constant cc such that if g(G)≥cg(G)≥c, then a′(G)≤(2+ε)Δ(G)+o(Δ(G))a′(G)≤(2+ε)Δ(G)+o(Δ(G)), where g(G)g(G) denotes the girth of GG.