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\(N\)-Flips in 4-Connected Even Triangulations on the Sphere

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Abstract

An even triangulation is a plane triangulation in which each vertex has even degree. It is well known that every even triangulation \(G\) has a unique 3-coloring, where the decomposition of the vertices of \(G\) by the 3-coloring is called the tripartition of \(G\). Nakamoto et al. (Graph Theory 51:260–268, 2006) proved that every two even triangulations with the same tripartition can be transformed into each other by \(N\)-flips, where an \(N\) -flip is an operation transforming an even triangulation into an even triangulation. In this paper, we prove that every two 4-connected even triangulations \(G\) and \(G'\) with the same tripartition can be transformed into each other by \(N\)-flips unless \(G\) or \(G'\) is isomorphic to a generalized octahedron.

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Acknowledgments

I am grateful to the two anonymous referees for carefully reading the paper and giving us helpful suggestions to improve the presentation of the paper.

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Authors and Affiliations

  1. Graduate School of Environment and Information Sciences, Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama, 240-8501, Japan

    Yuki Kawasaki & Atsuhiro Nakamoto

  2. Department of Computer and Information Science, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musashino-shi, Tokyo, 180-8633, Japan

    Naoki Matsumoto

Authors
  1. Yuki Kawasaki
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  2. Naoki Matsumoto
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  3. Atsuhiro Nakamoto
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Correspondence to Naoki Matsumoto.

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Kawasaki, Y., Matsumoto, N. & Nakamoto, A. \(N\)-Flips in 4-Connected Even Triangulations on the Sphere. Graphs and Combinatorics 31, 1889–1904 (2015). https://doi.org/10.1007/s00373-015-1574-x

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  • DOI: https://doi.org/10.1007/s00373-015-1574-x

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