Chromatic-choosability of the power of graphs

Abstract

The kth power Gk of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in Gk if the distance between u and v in G is at most k. Let χ(H) and χℓ(H) be the chromatic number and the list chromatic number of H, respectively. A graph H is called chromatic-choosable if χℓ(H)=χ(H). It is an interesting problem to find graphs that are chromatic-choosable. A natural question raised by Xuding Zhu (2013) is whether there exists a constant integer k such that Gk is chromatic-choosable for every graph G.Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) asked whether G2 is chromatic-choosable for every graph G. Kim and Park (2014) solved Kostochka and Woodall’s conjecture in the negative by finding a family of graphs G whose squares are complete multipartite graphs with partite sets of unbounded size. In this paper, we answer Zhu’s question by showing that for every integer k≥2, there exists a graph G such that Gk is not chromatic-choosable. Moreover, for any fixed k we show that the value χℓ(Gk)−χ(Gk) can be arbitrarily large.