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