A new upper bound on the acyclic chromatic indices of planar graphs

Abstract

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a^{\prime }(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. It was conjectured that $a^{\prime }(G)\le \Delta +2$ for any simple graph $G$ with maximum degree $\Delta$. In this paper, we prove that if $G$ is a planar graph, then $a^{\prime }(G)\le \Delta +7$. This improves a result by Basavaraju et al. [{\em Acyclic edge-coloring of planar graphs}, SIAM J. Discrete Math., 25 (2011), pp. 463-478], which says that every planar graph $G$ satisfies $a^{\prime }(G)\le \Delta +12$.

23 pages, 1 figures