Injective edge-coloring of graphs with small maximum degree

Abstract

An injective $k$-edge-coloring of a graph $G$ is a mapping $\ddot{I}•$: $E(G)\to \{1,2,...,k\}$, such that $\ddot{I}•(e)\ne \ddot{I}•(e^{\prime })$ if edges $e$ and $e^{\prime }$ are at distance two, or are in a triangle. The smallest integer $k$ such that $G$ has an injective $k$-edge-coloring is called the injective chromatic index of $G$, denoted by $\ddot{I}{‡}_i^{\prime }(G)$. In this paper, we prove that $\ddot{I}{‡}_i^{\prime }(G)\le 7$ for every graph $G$ with $\hat{I}”(G)\le 4$ and mad$(G)<\frac{8}{3}$, where $\hat{I}”(G)$ is the maximum degree of $G$.