TY - JOUR

T1 - Diameter of colorings under Kempe changes

AU - Bonamy, Marthe

AU - Heinrich, Marc

AU - Ito, Takehiro

AU - Kobayashi, Yusuke

AU - Mizuta, Haruka

AU - Mühlenthaler, Moritz

AU - Suzuki, Akira

AU - Wasa, Kunihiro

PY - 2020/10/24

Y1 - 2020/10/24

N2 - 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.

AB - 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.

KW - Combinatorial reconfiguration

KW - Gaph coloring

KW - Graph algorithms

KW - Kempe equivalence

UR - http://www.scopus.com/inward/record.url?scp=85086135132&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85086135132&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2020.05.033

DO - 10.1016/j.tcs.2020.05.033

M3 - Article

AN - SCOPUS:85086135132

VL - 838

SP - 45

EP - 57

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -