Chromatic-choosability of the power of graphs

TL;DR

This paper investigates the chromatic-choosability of graph powers, demonstrating that for any fixed power, there exist graphs where the list chromatic number significantly exceeds the chromatic number, answering a longstanding open question.

Contribution

It proves that for every integer k ≥ 2, there exists a graph G such that G^k is not chromatic-choosable, and the difference between list chromatic and chromatic numbers can be arbitrarily large.

Findings

For all k ≥ 2, G^k can be non-chromatic-choosable.

The difference χ_l(G^k) - χ(G^k) can be made arbitrarily large.

Answered Zhu's question negatively by constructing such graphs.

Abstract

The $k$th power $G^k$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^k$ if the distance between $u$ and $v$ in $G$ is at most $k$. Let $\chi(H)$ and $\chi_l(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $\chi_l (H) = \chi(H)$. It is an interesting problem to find graphs that are chromatic-choosable. A natural question raised by Xuding Zhu (2012) is whether there exists a constant integer $k$ such that $G^k$ is chromatic-choosable for every graph $G$. Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) asked whether $G^2$ is chromatic-choosable for every graph $G$. Kim and Park (2013) answered the Kostochka and Woodall's question in the negative by finding a family of graphs whose squares are complete multipartite graphs with partite sets of equal and unbounded size. In this paper, we answer Zhu's question by showing that for every integer $k \geq 2$, there exists a graph $G$ such that $G^k$ is not chromatic-choosable. Moreover, for any fixed $k$ we show that the value $\chi_l(G^k) - \chi(G^k)$ can be arbitrarily large.