Proper edge coloring with rainbow diamonds

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 ${\chi }_D^{\mathrm{\prime }}(G)$, is the minimum number of colors needed for a D-coloring of $G$. Denote by $\mathrm{\Delta }$ the maximum degree of $G$. We prove that ${\chi }_D^{\mathrm{\prime }}(G)\le \frac{9}{16}{\mathrm{\Delta }}^2+\frac{1}{2}\mathrm{\Delta }$, conjecture that ${\chi }_D^{\mathrm{\prime }}(G)\le \frac{1}{2}{\mathrm{\Delta }}^2+\frac{1}{2}\mathrm{\Delta }$, and verify this conjecture for $\mathrm{\Delta }\le 5$.