Injective edge chromatic index of sparse graphs

Abstract

An injective $k$-edge coloring of a graph $G$ is a $k$-edge coloring $\varphi$ of $G$ such that $\varphi \left(e_1\right)\ne \varphi \left(e_3\right)$ for any three consecutive edges $e_1,e_2$ and $e_3$ of a path or a triangle. The injective edge chromatic index ${\chi }_i^{\prime }\left(G\right)$ of $G$ is the smallest value of $k$ such that $G$ has an injective $k$-edge coloring. Let $mad\left(G\right)$ and $\Delta \left(G\right)$ denote the maximum average degree and the maximum degree of a graph $G$, respectively. In this paper we show that if $\Delta \left(G\right)\ge 3$ and $mad\left(G\right)<\frac{7}{3}$, then ${\chi }_i^{\prime }\left(G\right)\le \Delta \left(G\right)+1$; and if $\Delta \left(G\right)\ge 3$ and $mad\left(G\right)<\frac{5}{2}$, then ${\chi }_i^{\prime }\left(G\right)\le \Delta \left(G\right)+2$.