Further result on acyclic chromatic index of planar graphs

Abstract

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge coloring of $G$. It was conjectured that $\chiup'_{a}(G)\leq \Delta(G) + 2$ for any simple graph $G$ with maximum degree $\Delta(G)$. In this paper, we prove that every planar graph $G$ admits an acyclic edge coloring with $\Delta(G) + 6$ colors.