Kempe Equivalent List Colorings

Abstract

An $\alpha ,\beta$-Kempe swap in a properly colored graph interchanges the colors on some component of the subgraph induced by colors $\alpha$ and $\beta$. Two k-colorings of a graph are k-Kempe equivalent if we can form one from the other by a sequence of Kempe swaps (never using more than k colors). Las Vergnas and Meyniel showed that if a graph is $(k-1)$-degenerate, then each pair of its k-colorings are k-Kempe equivalent. Mohar conjectured the same conclusion for connected k-regular graphs. This was proved for $k=3$ by Feghali, Johnson, and Paulusma (with a single exception $K_2\square \,K_3$, also called the 3-prism) and for $k\ge 4$ by Bonamy, Bousquet, Feghali, and Johnson. In this paper we prove an analogous result for list-coloring. For a list-assignment L and an L-coloring $\varphi$, a Kempe swap is called L-valid for $\varphi$ if performing the Kempe swap yields another L-coloring. Two L-colorings are called L-equivalent if we can form one from the other by a sequence of L-valid Kempe swaps. Let G be a connected k-regular graph with $k\ge 3$ and $G\ne K_{k+1}$. We prove that if L is a k-assignment, then all L-colorings are L-equivalent (again excluding only $K_2\square \,K_3$). Further, if $G\in \{K_{k+1},K_2\square \,K_3\}$, L is a $\Delta$-assignment, but L is not identical everywhere, then all L-colorings of G are L-equivalent. When $k\ge 4$, the proof is completely self-contained, implying an alternate proof of the result of Bonamy et al. Our proofs rely on the following key lemma, which may be of independent interest. Let H be a graph such that for every degree-assignment $L_H$ all $L_H$-colorings are $L_H$-equivalent. If G is a connected graph that contains H as an induced subgraph, then for every degree-assignment $L_G$ for G all $L_G$-colorings are $L_G$-equivalent.