Proper edge coloring with rainbow diamonds

Abstract

Motivated by the B-coloring defined by Gyárfás and Sárközy, we introduce a new edge coloring called \emph{D-coloring}. For a graph $G$, a D-coloring of $G$ is a proper edge coloring such that every diamond subgraph is rainbow. The \emph{D-chromatic index} of $G$, denoted by $χ'_D(G)$, is the minimum number of colors needed for a D-coloring of $G$. Denote by $Δ$ the maximum degree of $G$. We prove that $χ'_D(G)\le \frac{9}{16}Δ^2+\frac{1}{2}Δ$, conjecture that $χ'_D(G)\le \frac{1}{2}Δ^2+\frac{1}{2}Δ$, and verify this conjecture for $Δ\le 5$.