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Rhombus Tilings of an Even-Sided Polygon and Quadrangulations on the Projective Plane

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Abstract

A quadrangulation on a closed surface is k-minimal if its shortest noncontractible cycle is of length k and if any face contraction yields a noncontractible cycle of length less than k. We prove that the rhombus tilings of a regular 2k-gon bijectively correspond to the pairs of a k-minimal quadrangulations on the projective plane and its specified noncontractible k-cycle.

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Acknowledgements

We are grateful to the anonymous referee for his valuable suggestion to introduce the notion of zonotopes. Using this notion, we have been able to explain our inductive proof of the results very naturally, and consider a possible extension of our result to 3-dimensional objects. These improvements along them has greatly enhanced the readability of the paper.

Funding

This research was supported by Japan Society for the Promotion of Science (KAKENHI Grant Number 18K03390).

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Authors and Affiliations

  1. Department of Mathematics and Natural Sciences, Hyogo University of Teacher Education, 942-1 Shimokume, Kato, Hyogo, 673-1494, Japan

    Hiroaki Hamanaka

  2. Faculty of Environment and Information Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-Ku, Yokohama, 240-8501, Japan

    Atsuhiro Nakamoto

  3. Department of Mathematics, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-Ku, Niigata, 950-2181, Japan

    Yusuke Suzuki

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  1. Hiroaki Hamanaka
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  2. Atsuhiro Nakamoto
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  3. Yusuke Suzuki
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Correspondence to Atsuhiro Nakamoto.

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Hamanaka, H., Nakamoto, A. & Suzuki, Y. Rhombus Tilings of an Even-Sided Polygon and Quadrangulations on the Projective Plane. Graphs and Combinatorics 36, 561–571 (2020). https://doi.org/10.1007/s00373-020-02137-0

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  • DOI: https://doi.org/10.1007/s00373-020-02137-0

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