On injective edge-coloring of graphs with maximum degree 4

Abstract

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 $G$, ${\chi }_{inj}^{\prime }\left(G\right)$, is the minimum number of colors needed for an injective edge-coloring of $G$. In this paper, we prove that if $G$ is graph with $\Delta \left(G\right)=4$ and maximum average degree is less than $\frac{5}{2}$ (resp. $\frac{13}{5}$, $\frac{36}{13}$), then ${\chi }_{inj}^{\prime }\left(G\right)\le 6$ (resp. 7, 8).