Injective edge-coloring of graphs with small maximum degree

An injective [Formula: see text]-edge-coloring of a graph [Formula: see text] is a mapping [Formula: see text]: [Formula: see text], such that [Formula: see text] if edges [Formula: see text] and [Formula: see text] are at distance two, or are in a triangle. The smallest integer [Formula: see text] such that [Formula: see text] has an injective [Formula: see text]-edge-coloring is called the injective chromatic index of [Formula: see text], denoted by [Formula: see text]. In this paper, we prove that [Formula: see text] for every graph [Formula: see text] with [Formula: see text] and mad[Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text].