Injective edge coloring of some sparse graphs.

A k-edge coloring φ of a graph G is injective if φ (e 1) ≠ φ (e 3) for any three consecutive edges e 1 , e 2 and e 3 in the same path or triangle. The injective chromatic index χ 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 ⊆ G } . We prove that every subcubic graph G has χ 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 χ i ′ (G) ≤ 12 if mad (G) < 33 10 , which improves the result of Miao et al. (Discrete Appl Math 310:65–74, 2022). [ABSTRACT FROM AUTHOR]