Graph r-hued colorings—A survey

Abstract

A ( k , r ) -coloring of a graph G is a proper k -vertex coloring of G such that the neighbors of each vertex of degree d will receive at least min { d , r } different colors. The r -hued chromatic number , denoted by χ r ( G ) , is the smallest integer k for which a graph G has a ( k , r ) -coloring. This article is intended to survey the recent developments on the studies related to this r -hued colorings. Emphases are on the r -hued colorings of planar graphs, graph families with forbidden minors, and sparse graphs, as well as on the comparison between the r -hued chromatic number and the chromatic number of a graph, and the sensitivity studies of the r -hued chromatic number. It also surveys other related results on r -hued colorings and list r -hued colorings.