An improved bound on acyclic chromatic index of planar graphs

Abstract

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 ${\chi }_a^{\prime }(G)$, is the least number of colors $k$ such that $G$ has an acyclic edge $k$-coloring. Basavaraju et al. [Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2) (2011), 463--478] showed that ${\chi }_a^{\prime }(G)\le \Delta (G)+12$ for planar graphs $G$ with maximum degree $\Delta (G)$. In this paper, the bound is improved to $\Delta (G)+10$.