Acyclic Edge Coloring of Triangle-Free Planar Graphs

Abstract

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) ⩽ Δ + 2, where Δ = Δ(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 article, we prove that if G satisfies Property A, then a′(G) ⩽ Δ + 3. Triangle-free planar graphs satisfy Property A. We infer that a′(G) ⩽ Δ + 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). © 2011 Wiley Periodicals, Inc. J Graph Theory