Kempe Equivalent List Colorings Revisited

TL;DR

This paper proves that for every 4-connected, non-complete graph and any degree-assignment, all list colorings are connected through valid Kempe changes, extending understanding of coloring equivalence.

Contribution

It establishes that all list colorings are equivalent via Kempe changes for 4-connected, non-complete graphs under any degree-assignment.

Findings

All $L$-colorings are $L$-equivalent for the specified class of graphs.

Extends Kempe chain techniques to broader classes of list colorings.

Addresses a recent question by Cranston and Mahmoud.

Abstract

A \emph{Kempe chain} on colors $a$ and $b$ is a component of the subgraph induced by colors $a$ and $b$. A \emph{Kempe change} is the operation of interchanging the colors of some Kempe chain. For a list-assignment $L$ and an $L$-coloring $\varphi$, a Kempe change is \emph{$L$-valid} for $\varphi$ if performing the Kempe change yields another $L$-coloring. Two $L$-colorings are \emph{$L$-equivalent} if we can form one from the other by a sequence of $L$-valid Kempe changes. A \emph{degree-assignment} is a list-assignment $L$ such that $L(v)\ge d(v)$ for every $v\in V(G)$. Cranston and Mahmoud (\emph{Combinatorica}, 2023) asked: For which graphs $G$ and degree-assignment $L$ of $G$ is it true that all the $L$-colorings of $G$ are $L$-equivalent? We prove that for every 4-connected graph $G$ which is not complete and every degree-assignment $L$ of $G$, all $L$-colorings of $G$ are $L$-equivalent.