1. Diameter of colorings under Kempe changes.
- Author
-
Bonamy, Marthe, Heinrich, Marc, Ito, Takehiro, Kobayashi, Yusuke, Mizuta, Haruka, Mühlenthaler, Moritz, Suzuki, Akira, and Wasa, Kunihiro
- Subjects
- *
PLANAR graphs , *GRAPH coloring , *GRAPH algorithms , *BIPARTITE graphs , *COLORS , *DIAMETER - Abstract
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. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF