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Reconfiguration of Colorings in Triangulations of the Sphere

Authors Takehiro Ito , Yuni Iwamasa , Yusuke Kobayashi , Shun-ichi Maezawa , Yuta Nozaki , Yoshio Okamoto , Kenta Ozeki

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Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Yuni Iwamasa
  • Graduate School of Informatics, Kyoto University, Japan
Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Shun-ichi Maezawa
  • Department of Mathematics, Tokyo University of Science, Japan
Yuta Nozaki
  • Faculty of Environment and Information Sciences, Yokohama National University, Japan
  • SKCM², Hiroshima University, Japan
Yoshio Okamoto
  • Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo, Japan
Kenta Ozeki
  • Faculty of Environment and Information Sciences, Yokohama National University, Japan


We thank Naonori Kakimura and Naoyuki Kamiyama for related discussion, and anonymous reviewers for helpful suggestions.

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Takehiro Ito, Yuni Iwamasa, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Yoshio Okamoto, and Kenta Ozeki. Reconfiguration of Colorings in Triangulations of the Sphere. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 43:1-43:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph coloring
  • Graph coloring
  • Triangulation of the sphere
  • Combinatorial reconfiguration


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