A stronger upper bound on the D-chromatic index

Abstract

For a graph $G$, a proper edge coloring of $G$ is called a D-coloring if every diamond subgraph of $G$ is rainbow. Let $χ'_D(G)$ be the D-chromatic index of $G$, which is the smallest integer $k$ such that $G$ admits a D-coloring with $k$ colors. Let $Δ$ be the maximum degree of $G$. The only known Brooks-type upper bound on $χ'_D(G)$ is $\frac{9}{16}Δ^2 + \frac{1}{2}Δ$, given by a greedy coloring. In this paper, using a probabilistic method, we obtain the first improvement upon this upper bound by proving that $χ'_D(G) \le (1-c)\frac{9}{16}Δ^2$ for some $c > 0$ and sufficiently large $Δ$.