On injective chromatic index of sparse graphs with maximum degree 5

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 \(e_3\) of a path or a triangle. The injective chromatic index \(\chi _i'(G)\) 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 \(\Delta \le 5\) has \(\chi _i'(G)\le 12\) (resp. 13) if its maximum average degree is less than \(\frac{20}{7}\) (resp. 3), which improves the results of Zhu (2023).