Remark on injective edge-coloring of some sparse graphs

Abstract

A $k$-edge coloring of a graph $G$ is considered injective if any two edges that are at distance 2 or reside within the same triangle receive distinct colors. The minimum integer $k$ for which $G$ admits a $k$-injective-edge coloring is referred to as the injective edge chromatic number of $G$, denoted by $\chi_i^{'}(G)$. This paper presents findings on the injective edge chromatic numbers for graphs with maximum degrees of 4 and 5. In particular, we show that if the maximum average degree ($mad(G)$) of a graph $G$, where the maximum degree is $4$, is less than $\frac{38}{11}$, then it follows that $\chi_i^{'}(G)\leq13$, thereby enhancing the previous result established by Bu and Qi [Discrete Math. Algorithms Appl., 2018]. Additionally, we prove that for any graph $G$ with a maximum degree of $5$, it holds true that if $mad(G)<\frac{1501}{384}$, then $\chi_i^{'}(G)\leq21$, furthermore, if $mad(G)<4$, then also $\chi_i^{'}(G)\leq22$.