Acyclic edge coloring of sparse graphs

A proper edge coloring of a graph G is called acyclic if there is no bichromatic cycle in G. The acyclic chromatic index of G, denoted by χa(G), is the least number of colors k such that G has an acyclic edge k-coloring. The maximum average degree of a graph G, denoted by mad(G), is the maximum of the average degree of all subgraphs of G. In this paper, it is proved that if mad(G) < 4, then χa(G) ≤ ∆(G) + 2; if mad(G) < 3, then χ ′ a(G) ≤ ∆(G) + 1. This implies that every triangle-free planar graph G is acyclically edge (∆(G) + 2)-colorable.