Kempe equivalent list colorings revisited

A Kempe chain on colors a and b is a component of the subgraph induced by colors a and b. A Kempe change is the operation of interchanging the colors of some Kempe chains. For a list‐assignment L and an L‐coloring φ, a Kempe change is L‐valid for φ if performing the Kempe change yields another L‐coloring. Two L‐colorings are L‐equivalent if we can form one from the other by a sequence of L‐valid Kempe changes. A degree‐assignment is a list‐assignment L such that Lv dv () () ≥ for every v VG ∈ ( ). Cranston and Mahmoud 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.