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

Abstract

An injective $k$-edge-coloring of a graph $G$ is a mapping $\varphi$: $E(G)\to \{1,2,...,k\}$, such that $\varphi (e)\ne \varphi (e^{\prime })$ 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 ${\chi }_i^{\prime }(G)$. 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 ${\chi }_i^{\prime }(G)\le 13$ for every claw-free graph $G$ with $\Delta (G)\le 4$, where $\Delta (G)$ is the maximum degree of $G$.