2-Distance Coloring of Planar Graphs with Girth at Least 4

Abstract

A [Formula: see text]-distance [Formula: see text]-coloring of a graph is a coloring of the vertices with [Formula: see text] colors in which any two vertices at distance at most [Formula: see text] receive distinct colors. The [Formula: see text]-distance chromatic number of [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] admits a [Formula: see text]-distance [Formula: see text]-coloring. The girth of a graph [Formula: see text], denoted by [Formula: see text], is the length of its shortest cycle. We show that if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text], then [Formula: see text]. Moreover, we prove that if [Formula: see text] is a planar graph with [Formula: see text] and [Formula: see text], then [Formula: see text].