TY - GEN

T1 - Diameter of Colorings Under Kempe Changes

AU - Bonamy, Marthe

AU - Heinrich, Marc

AU - Ito, Takehiro

AU - Kobayashi, Yusuke

AU - Mizuta, Haruka

AU - MÃ¼hlenthaler, Moritz

AU - Suzuki, Akira

AU - Wasa, Kunihiro

N1 - Funding Information:
Supported partially by JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA). Also supported partially by the ANR Project GrR (ANR-18-CE40-0032) operated by the French National Research Agency (ANR), by JSPS KAKENHI Grant Numbers JP16H03118, JP16K16010, JP17K12636, JP17K19960, JP18H04091, JP18H05291, JP19J10042, JP19K11814, and JP19K20350, Japan, and by JST CREST Grant Numbers JPMJCR1401, JPMJCR1402, and JPMJCR18K3, Japan.
Publisher Copyright:
© Springer Nature Switzerland AG 2019.

PY - 2019

Y1 - 2019

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 ≥, 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 ≥, 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.

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U2 - 10.1007/978-3-030-26176-4_5

DO - 10.1007/978-3-030-26176-4_5

M3 - Conference contribution

AN - SCOPUS:85070200593

SN - 9783030261757

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 52

EP - 64

BT - Computing and Combinatorics - 25th International Conference, COCOON 2019, Proceedings

A2 - Du, Ding-Zhu

A2 - Duan, Zhenhua

A2 - Tian, Cong

PB - Springer Verlag

T2 - 25th International Computing and Combinatorics Conference, COCOON 2019

Y2 - 29 July 2019 through 31 July 2019

ER -