Injective edge coloring of some sparse graphs

Abstract

A k-edge coloring $\varphi$ of a graph G is injective if $\varphi (e_1)\ne \varphi (e_3)$ for any three consecutive edges $e_1, e_2$ and $e_3$ in the same path or triangle. The injective chromatic index $\chi _i'(G)$ is the smallest k necessary for an injective k-edge coloring of G. Let ${\text {mad}}(G)=\max \{\frac{2|E(H)|}{|V(H)|}:H\subseteq G\}$. We prove that every subcubic graph G has $\chi _i'(G)\le 6$ if ${\text {mad}}(G)<\frac{30}{11}$, which improves the result of Ferdjallah et al. (Injective edge-coloring of sparse graphs, 2020). We also prove that every graph G with maximum degree 4 has $\chi _i'(G)\le 12$ if ${\text {mad}}(G)<\frac{33}{10}$, which improves the result of Miao et al. (Discrete Appl Math 310:65–74, 2022).