Injective edge-coloring of claw-free graphs with maximum degree 4

An injective $k$-edge-coloring of a graph $G$ is a mapping $\phi$: $E(G)\rightarrow\{1,2,...,k\}$, such that $\phi(e)\ne\phi(e')$ if edges $e$ and $e'$ 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 $\chi_i'(G)$. A graph is called claw-free if it has no induced subgraph isomorphic to the complete bipartite graph $K_{1,3}$. In this paper, we show that $\chi_i'(G)\le 13$ for every claw-free graph $G$ with $\Delta(G)\leq 4$, where $\Delta(G)$ is the maximum degree of $G$.