Diameter of colorings under Kempe changes

Marthe Bonamy, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Moritz Mühlenthaler, Akira Suzuki, Kunihiro Wasa

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


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≥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.

Original languageEnglish
Pages (from-to)45-57
Number of pages13
JournalTheoretical Computer Science
Publication statusPublished - 2020 Oct 24


  • Combinatorial reconfiguration
  • Gaph coloring
  • Graph algorithms
  • Kempe equivalence


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