Acyclic Edge Coloring of Triangle Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a(G) ≤ ∆ + 2, where ∆ = ∆(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ≤ 2|V (H)| − 1, we say that the graph G satisfies Property A. In this paper, we prove that if G satisfies Property A, then a(G) ≤ ∆ + 3. Triangle free planar graphs satisfy Property A. We infer that a(G) ≤ ∆ + 3, if G is a triangle free planar graph. Another class of graph which satisfies Property A is 2-fold graphs (union of two forests).