Counterexamples to the List Square Coloring Conjecture

TL;DR

This paper presents infinitely many counterexamples to the List Square Coloring Conjecture, demonstrating that the list chromatic number of a graph's square can significantly exceed its chromatic number.

Contribution

The authors construct explicit counterexamples to the conjecture, showing that the list chromatic number of the square of a graph can be arbitrarily larger than its chromatic number.

Findings

Counterexamples to the List Square Coloring Conjecture are infinite.

The difference between list chromatic number and chromatic number can be arbitrarily large.

The conjecture does not hold universally for all graphs.

Abstract

The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. 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. Kostochka and Woodall \cite{KW2001} conjectured that $\chi_l(G^2) = \chi(G^2)$ for every graph $G$, which is called List Square Coloring Conjecture. In this paper, we give infinitely many counterexamples to the conjecture. Moreover, we show that the value $\chi_l(G^2) - \chi(G^2)$ can be arbitrary large.