Diameter of Colorings Under Kempe Changes

Given a k-coloring of a graph G, a Kempe-change for two colors a and b produces another k-coloring of G, as follows: first choose a connected component in the subgraph of G induced by the two color classes of a and b, and then swap the colors a and b in the component. Two k-colorings are called Kempe-equivalent if one can be transformed into the other by a sequence of Kempe-changes. We consider two problems, defined as follows: First, given two k-colorings of a graph G, Kempe Reachability asks whether they are Kempe-equivalent; and second, given a graph G and a positive integer k, Kempe Connectivity asks whether any two k-colorings of G are Kempe-equivalent. We analyze the complexity of these problems from the viewpoint of graph classes. We prove that Kempe Reachability is PSPACE-complete for any fixed \(k \ge 3\), and that it remains PSPACE-complete even when restricted to three colors and planar graphs of maximum degree six. Furthermore, we show that both problems admit polynomial-time algorithms on chordal graphs, bipartite graphs, and cographs. For each of these graph classes, we give a non-trivial upper bound on the number of Kempe-changes needed in order to certify that two k-colorings are Kempe-equivalent.