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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.43
URN: urn:nbn:de:0030-drops-178936
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17893/
Ito, Takehiro ;
Iwamasa, Yuni ;
Kobayashi, Yusuke ;
Maezawa, Shun-ichi ;
Nozaki, Yuta ;
Okamoto, Yoshio ;
Ozeki, Kenta
Reconfiguration of Colorings in Triangulations of the Sphere
Abstract
In 1973, Fisk proved that any 4-coloring of a 3-colorable triangulation of the 2-sphere can be obtained from any 3-coloring by a sequence of Kempe-changes. On the other hand, in the case where we are only allowed to recolor a single vertex in each step, which is a special case of a Kempe-change, there exists a 4-coloring that cannot be obtained from any 3-coloring.
In this paper, we present a linear-time checkable characterization of a 4-coloring of a 3-colorable triangulation of the 2-sphere that can be obtained from a 3-coloring by a sequence of recoloring operations at single vertices. In addition, we develop a quadratic-time algorithm to find such a recoloring sequence if it exists; our proof implies that we can always obtain a quadratic length recoloring sequence. We also present a linear-time checkable criterion for a 3-colorable triangulation of the 2-sphere that all 4-colorings can be obtained from a 3-coloring by such a sequence. Moreover, we consider a high-dimensional setting. As a natural generalization of our first result, we obtain a polynomial-time checkable characterization of a k-coloring of a (k-1)-colorable triangulation of the (k-2)-sphere that can be obtained from a (k-1)-coloring by a sequence of recoloring operations at single vertices and the corresponding algorithmic result. Furthermore, we show that the problem of deciding whether, for given two (k+1)-colorings of a (k-1)-colorable triangulation of the (k-2)-sphere, one can be obtained from the other by such a sequence is PSPACE-complete for any fixed k ≥ 4. Our results above can be rephrased as new results on the computational problems named k-Recoloring and Connectedness of k-Coloring Reconfiguration Graph, which are fundamental problems in the field of combinatorial reconfiguration.
BibTeX - Entry
@InProceedings{ito_et_al:LIPIcs.SoCG.2023.43,
author = {Ito, Takehiro and Iwamasa, Yuni and Kobayashi, Yusuke and Maezawa, Shun-ichi and Nozaki, Yuta and Okamoto, Yoshio and Ozeki, Kenta},
title = {{Reconfiguration of Colorings in Triangulations of the Sphere}},
booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)},
pages = {43:1--43:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-273-0},
ISSN = {1868-8969},
year = {2023},
volume = {258},
editor = {Chambers, Erin W. and Gudmundsson, Joachim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17893},
URN = {urn:nbn:de:0030-drops-178936},
doi = {10.4230/LIPIcs.SoCG.2023.43},
annote = {Keywords: Graph coloring, Triangulation of the sphere, Combinatorial reconfiguration}
}
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Keywords: |
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Graph coloring, Triangulation of the sphere, Combinatorial reconfiguration |
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Collection: |
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39th International Symposium on Computational Geometry (SoCG 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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09.06.2023 |
