Injective-edge-coloring of planar graphs with girth restriction

Abstract

A [Formula: see text]-[Formula: see text]-[Formula: see text]-[Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] [Formula: see text] is a mapping [Formula: see text] such that [Formula: see text] for any three consecutive edges [Formula: see text],[Formula: see text],[Formula: see text] of a path or a [Formula: see text]-cycle. [Formula: see text][Formula: see text] has a [Formula: see text]-injective-edge-coloring[Formula: see text] is called the injective chromatic index of G. In this paper, we prove that for planar graphs [Formula: see text] with [Formula: see text], (1)[Formula: see text] if [Formula: see text]; (2)[Formula: see text] if [Formula: see text].