Injective Edge-Coloring of Graphs with Small Weight

Abstract

The edge weight, denoted by $\Delta _e(G)$, of a graph G is max$\{d_G(u)+d_G(v):uv\in E(G)\}$. An edge-coloring of a graph G is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance 2 in G or in a common triangle. The injective chromatic index of G, denoted by $\chi _i'(G)$, is the minimum number of colors needed for an injective edge-coloring of G. Ferdjallah, Kerdjoudj and Raspaud (Injective edge-coloring of sparse graphs, 2019, arXiv:1907.09838) posed a conjecture which says that if G is subcubic graph, then $\chi _i'(G) \le 6$. Kostochka, Raspaud and Xu (Eur J Combin 96:103355, 2021) showed that if G is a subcubic graph, then $\chi _i'(G) \le 7$. We extend the above result by showing that

$$\chi '_{i}(G)\le \left\{ \begin{array}{ll} 4, &{} \quad {\text { if } \Delta _e(G)\le 5,}\\ 7 , &{} \quad {\text { if } \Delta _e(G)\le 6,}\\ 12, &{} \quad {\text { if } \Delta _e(G)\le 7}. \end{array} \right. $$

In addition, we prove that $\chi _i'(G) \le 6$ for any subcubic claw-free graph, and this bound is sharp.