Injective edge-coloring of graphs with small maximum degree

An injective $k$-edge-coloring of a graph $G$ is a mapping $\phi$: $E(G)\to \{1,2,\mathrm{.}\mathrm{.}\mathrm{.},k\}$, such that $\phi (e)\mathrm{\ne }\phi (e^{\mathrm{\prime }})$ if edges $e$ and $e^{\mathrm{\prime }}$ 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 ${\chi }_i^{\mathrm{\prime }}(G)$. In this paper, we prove that ${\chi }_i^{\mathrm{\prime }}(G)\le 7$ for every graph $G$ with $\mathrm{\Delta }(G)\le 4$ and mad$(G)<\frac{8}{3}$, where $\mathrm{\Delta }(G)$ is the maximum degree of $G$.