Injective edge coloring of sparse graphs

A [Formula: see text]-edge coloring of a graph [Formula: see text] is a mapping [Formula: see text]. A [Formula: see text]-injective edge coloring is a [Formula: see text]-edge coloring such that any two edges [Formula: see text] and [Formula: see text] with distance exactly [Formula: see text] or in a triangle receive distinct colors. The injective edge chromatic index of a graph [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-injective edge coloring. In this paper, we show that [Formula: see text] if [Formula: see text]; [Formula: see text] if [Formula: see text]; and [Formula: see text] if [Formula: see text]. We also show that [Formula: see text] if [Formula: see text] is subcubic and [Formula: see text].