Every 4-regular graph is acyclically edge-6-colorable

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. Fiam${\rm \check{c}}$ik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that $a'(G)\le \Delta + 2$ for any simple graph $G$ with maximum degree $\Delta$. Basavaraju and Chandran (2009) showed that every graph $G$ with $\Delta=4$, which is not 4-regular, satisfies the conjecture. In this paper, we settle the 4-regular case, i.e., we show that every 4-regular graph $G$ has $a'(G)\le 6$.Comment: 24 pages, 9 figure

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

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. It was conjectured that $a'(G)\le \Delta+2$ for any simple graph $G$ with maximum degree $\Delta$. In this paper, we prove that if $G$ is a planar graph, then $a'(G)\leq\Delta +7$. 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 $a'(G)\leq\Delta +12$.Comment: 23 pages, 1 figure