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

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 ${\chi }_{inj}^{\mathrm{\prime }}(G)$, is the smallest integer $k$ such that $G$ admits an injective edge coloring with $k$ colors. Let $\mathrm{\Delta }$ be the maximum degree of $G$. Ferdjallah et al.~proved that ${\chi }_{inj}^{\mathrm{\prime }}(G)\le 2(\mathrm{\Delta }-1{)}^2$. In this paper, we improve this bound by proving that ${\chi }_{inj}^{\mathrm{\prime }}(G)\le 2(\mathrm{\Delta }-1{)}^2-\mathrm{\Delta }+3$ if $\mathrm{\Delta }\ge 4$.