Equitable coloring of planar graphs without 5-cycles and chordal 4-cycles

Abstract

An equitable [Formula: see text]-coloring of a graph [Formula: see text] is a proper vertex coloring such that [Formula: see text] for any [Formula: see text], where [Formula: see text] ([Formula: see text]) is the set of vertices colored with [Formula: see text]. If there is an equitable [Formula: see text]-coloring of [Formula: see text], then the graph [Formula: see text] is said to be equitably [Formula: see text]-colorable. In this paper, we prove that every planar graph without 5-cycles and chordal 4-cycles has an equitable [Formula: see text]-coloring for [Formula: see text] where [Formula: see text] is the maximum degree of [Formula: see text].