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The Matching Extendability of Optimal 1-Planar Graphs

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Abstract

A graph G is said to be 1-planar if it can be drawn on the sphere or plane so that any edge of G has at most one crossing point with another edge. Moreover, G is called an optimal 1-planar graph if \(|E(G)| = 4|V(G)|-8\). In this paper, we investigate the matching extendability of optimal 1-planar graphs. It is shown that every optimal 1-planar graph G of even order is 2-extendable unless G contains a 4-cycle C which separates the graph into two odd components. Moreover, for any 5-connected optimal 1-planar graph, we characterize a matching with three edges which is not extendable.

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Acknowledgements

The authors would like to thank Katsuhiro Ota whose comment led to significant improvement in the proof of Lemma 4.

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Authors and Affiliations

  1. Faculty of Business and Commerce, Keio University, Hiyoshi 4-1-1, Kohoku-ku, Yokohama, Kanagawa, 223-8521, Japan

    Jun Fujisawa

  2. Graduate School of Science and Technology, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan

    Keita Segawa

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

    Yusuke Suzuki

Authors
  1. Jun Fujisawa
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  2. Keita Segawa
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  3. Yusuke Suzuki
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Corresponding author

Correspondence to Jun Fujisawa.

Additional information

J. Fujisawa: Partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B) 16H03952 and (C) 17K05349 and Grant-in-Aid for Young Scientists (B) 26800085.

Y. Suzuki: Partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C) 16K05250.

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Fujisawa, J., Segawa, K. & Suzuki, Y. The Matching Extendability of Optimal 1-Planar Graphs. Graphs and Combinatorics 34, 1089–1099 (2018). https://doi.org/10.1007/s00373-018-1932-6

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  • DOI: https://doi.org/10.1007/s00373-018-1932-6

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