Further results on injective edge coloring of graphs with maximum degree 5

A graph $G$ is called injective $k$-edge colorable if it has a $k$-edge coloring $\phi$ such that any three consecutive edges $e_1,e_2$, and $e_3$ in the same path or triangle satisfy $\phi \left(e_1\right)\ne \phi \left(e_3\right)$. The injective chromatic index ${\chi }_i^{\prime }\left(G\right)$ is the smallest $k$ such that $G$ is injective $k$-edge colorable. We demonstrate that any graph with maximum degree 5 has ${\chi }_i^{\prime }\left(G\right)$ at most 7 (resp. 8, 9, 10) if its maximum average degree is less than $\frac{7}{3}$ (resp. $\frac{12}{5}$, $\frac{5}{2}$, $\frac{8}{3}$), which improves some results of Zhu et al. (2023) and Bu et al. (2024).