Injective edge coloring of sparse graphs with maximum degree 5

Abstract

A k-injective-edge coloring of a graph G is a mapping $c:E(G)\rightarrow \{1,2,\cdots ,k\}$ such that $c(e_1)\ne c(e_3)$ for any three consecutive edges $e_1,e_2,e_3$ of a path or a 3-cycle. $\chi _{i}'(G)=\min \{k|G$ has a k-injective-edge coloring$\}$ is called the injective chromatic index of G. In this paper, we prove that for graphs G with $\Delta (G)\le 5$, (1) $\chi _{i}'(G)\le 8$ if $mad(G)<\frac{7}{3}$; (2) $\chi _{i}'(G)\le 9$ if $mad(G)<\frac{12}{5}$; (3) $\chi _{i}'(G)\le 10$ if $mad(G)<\frac{5}{2}$; (4) $\chi _{i}'(G)\le 11$ if $mad(G)<\frac{18}{7}$.