On injective chromatic index of sparse graphs with maximum degree 5

A k-edge coloring \(\varphi \) $\phi$ of a graph G is injective if \(\varphi (e_1)\ne \varphi (e_3)\) $\phi (e_1)\ne \phi (e_3)$ for any three consecutive edges \(e_1, e_2\) $e_1,e_2$ and \(e_3\) $e_3$ of a path or a triangle. The injective chromatic index \(\chi _i'(G)\) ${\chi }_i^{\prime }(G)$ of G is the smallest k such that G admits an injective k-edge coloring. By discharging method, we demonstrate that any graph with maximum degree \(\Delta \le 5\) $\mathrm{\Delta }\le 5$ has \(\chi _i'(G)\le 12\) ${\chi }_i^{\prime }(G)\le 12$ (resp. 13) if its maximum average degree is less than \(\frac{20}{7}\) $\frac{20}{7}$ (resp. 3), which improves the results of Zhu (2023).