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 $χ'_{inj}(G)$, is the smallest integer $k$ such that $G$ admits an injective edge color