Kempe Equivalent List Colorings

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.