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Kempe Equivalence Classes of Cubic Graphs Embedded on the Projective Plane

Abstract

A Kempe switch of a 3-edge-coloring of a cubic graph G on a bicolored cycle C swaps the colors on C and gives rise to a new 3-edge-coloring of G. Two 3-edge-colorings of G are Kempe equivalent if they can be obtained from each other by a sequence of Kempe switches. Fisk proved that any two 3-edge-colorings in a cubic bipartite planar graph are Kempe equivalent. In this paper, we obtain an analog of this theorem and prove that all 3-edge-colorings of a cubic bipartite projective-planar graph G are pairwise Kempe equivalent if and only if G has an embedding in the projective plane such that the chromatic number of the dual triangulation G* is at least 5. As a by-product of the results in this paper, we prove that the list-edge-coloring conjecture holds for cubic graphs G embedded on the projective plane provided that the dual G* is not 4-vertex-colorable.

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Acknowledgment

The author thanks the anonymous reviewers for carefully reading the paper and for their helpful comments, which considerably improved both the content and the readability of the paper. The author is also grateful to Yuta Nozaki, who constructed cubic bipartite graphs embedded on the projective plane having more than three Kempe equivalence classes, as in Figure 13. This work was supported by JSPS KAKENHI, Grant Numbers 18K03391 and 20H05795, and the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.

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Correspondence to Kenta Ozeki.

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Dedicated to Professor Katsuhiro Ota on the occasion of his 60th birthday

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Ozeki, K. Kempe Equivalence Classes of Cubic Graphs Embedded on the Projective Plane. Combinatorica (2022). https://doi.org/10.1007/s00493-021-4330-2

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  • DOI: https://doi.org/10.1007/s00493-021-4330-2

Mathematics Subject Classification (2010)

  • 05C10
  • 05C15