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 K 2 □ K 3 , 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 K 2 □ K 3 ). Further, if G ∈ { K k + 1 , K 2 □ K 3 } , 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.