Injective edge coloring of some sparse graphs

A k-edge coloring phi of a graph G is injective if phi (e(1)) not equal phi(e(3)) for any three consecutive edges e(1), e(2) and e(3) in the same path or triangle. The injective chromatic index X'(i) (G) is the smallest k necessary for an injective k-edge coloring of G. Let mad(G) = max{2|E(H)|/|V(H)| : H subset of G}. We prove that every subcubic graph G has X'(i) (G) <= 6 if mad(G) < 30/11, which improves the result of Ferdjallah et al. (Injective edge-coloring of sparse graphs, 2020). We also prove that every graph G with maximum degree 4 has X'(i) (G) <= 12 if mad (G) < 33/10, which improves the result of Miao et al. (Discrete Appl Math 310:65-74, 2022).