Kempe Equivalent List Colorings

An α,β-Kempe swap in a properly colored graph interchanges the colors on some component of the subgraph induced by colors α and β. 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 K₂□K₃, also called the 3-prism) and for k≥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 φ, a Kempe swap is called L-valid for φ 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≥3 and G≠K_k+1. We prove that if L is a k-assignment, then all L-colorings are L-equivalent (again excluding only K₂□K₃). Further, if G∈{K_k+1,K₂□K₃}, L is a Δ-assignment, but L is not identical everywhere, then all L-colorings of G are L-equivalent. When k≥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.