(1,0)-Relaxed strong edge list coloring of planar graphs with girth 6

Abstract

Let [Formula: see text] be a graph. An [Formula: see text]-relaxed strong edge [Formula: see text]-coloring is a mapping [Formula: see text] such that for any edge [Formula: see text], there are at most [Formula: see text] edges adjacent to [Formula: see text] and [Formula: see text] edges which are distance two apart from [Formula: see text] assigned the same color as [Formula: see text]. The [Formula: see text]-relaxed strong chromatic index, denoted by [Formula: see text], is the minimum number [Formula: see text] of an [Formula: see text]-relaxed strong [Formula: see text]-edge-coloring admitted by [Formula: see text]. [Formula: see text] is called [Formula: see text]-relaxed strong edge [Formula: see text]-colorable if for a given list assignment [Formula: see text], there exists an [Formula: see text]-relaxed strong edge coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text]. If [Formula: see text] is [Formula: see text]-relaxed strong edge [Formula: see text]-colorable for any list assignment with [Formula: see text] for all [Formula: see text], then [Formula: see text] is said to be [Formula: see text]-relaxed strong edge [Formula: see text]-choosable. The [Formula: see text]-relaxed strong list chromatic index, denoted by [Formula: see text], is defined to be the smallest integer [Formula: see text] such that [Formula: see text] is [Formula: see text]-relaxed strong edge [Formula: see text]-choosable. In this paper, we prove that every planar graph [Formula: see text] with girth 6 satisfies that [Formula: see text]. This strengthens a result which says that every planar graph [Formula: see text] with girth 7 and [Formula: see text] satisfies that [Formula: see text].