Remark on injective edge-coloring of some sparse graphs

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 χ_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 38/11 , then it follows that χ_i'(G) ≤ 13 , 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) < 1501/384, then χ_i'(G) ≤ 21 , furthermore, if mad(G) < 4, then also χ_i'(G) ≤ 22 .