A note on the injective edge coloring of graphs with bounded maximum degree

Abstract

Let $G$ be a graph. Under an \emph{injective edge coloring} of $G$, any two edges at distance $2$ or belonging to a common triangle receive distinct colors. The \emph{injective chromatic index} of $G$, denoted by ${\chi }_{inj}^{\prime }(G)$, is the smallest integer $k$ such that $G$ admits an injective edge coloring with $k$ colors. Let $\Delta$ be the maximum degree of $G$. Ferdjallah et al.~proved that ${\chi }_{inj}^{\prime }(G)\le 2(\Delta -1{)}^2$. In this paper, we improve this bound by proving that ${\chi }_{inj}^{\prime }(G)\le 2(\Delta -1{)}^2-\Delta +3$ if $\Delta \ge 4$.