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Japanese Conference on Discrete and Computational Geometry and Graphs

JCDCGG 2013: Discrete and Computational Geometry and Graphs pp 96–111Cite as

Properly Colored Geometric Matchings and 3-Trees Without Crossings on Multicolored Points in the Plane

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8845))

Abstract

Let \(X\) be a set of multicolored points in the plane such that no three points are collinear and each color appears on at most \(\lceil |X|/2 \rceil \) points. We show the existence of a non-crossing properly colored geometric perfect matching on \(X\) (if \(|X|\) is even), and the existence of a non-crossing properly colored geometric spanning tree with maximum degree at most \(3\) on \(X\). Moreover, we show the existence of a non-crossing properly colored geometric perfect matching in the plane lattice. In order to prove these our results, we propose an useful lemma that gives a good partition of a sequence of multicolored points.

M. Kano—Partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C).

K. Suzuki—Partially supported by MEXT. KAKENHI 24740068.

52C35: Arrangements of points, flats, hyperplanes. 05C70: Factorization, matching, partitioning, covering and packing. 05C05: Trees.

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Notes

  1. 1.

    We denote a set of black points by \(K\) not by \(B\), because \(B\) means a set of blue points in this paper.

References

  1. Hoffmann, M., Tóth, C.D.: Vertex-colored encompassing graphs. Graphs and Combinatorics 30, 933–947 (2014)

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  2. Kaneko, A.: On the maximum degree of bipartite embeddings of trees in the plane. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 166–171. Springer, Heidelberg (2000)

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  3. Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane — a survey —. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry. Algorithms and Combinatorics, vol. 25, pp. 551–570. Springer, Heidelberg (2003)

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  4. Kano, M., Suzuki, K.: Discrete geometry on red and blue points in the plane lattice. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 355–369. Springer, New York (2013)

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Author information

Authors and Affiliations

  1. Department of Computer and Information Sciences, Ibaraki University, Hitachi, Ibaraki, Japan

    Mikio Kano

  2. Department of Information Science, Kochi University, Kochi, Japan

    Kazuhiro Suzuki

  3. University Education Center, Ibaraki University, Mito, Ibaraki, Japan

    Miyuki Uno

Authors
  1. Mikio Kano
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  2. Kazuhiro Suzuki
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  3. Miyuki Uno
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Corresponding author

Correspondence to Kazuhiro Suzuki .

Editor information

Editors and Affiliations

  1. Tokyo University of Science, Tokyo, Japan

    Jin Akiyama

  2. The University of Electro-Communications, Tokyo, Japan

    Hiro Ito

  3. Tokai University, Tokyo, Japan

    Toshinori Sakai

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© 2014 Springer International Publishing Switzerland

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Kano, M., Suzuki, K., Uno, M. (2014). Properly Colored Geometric Matchings and 3-Trees Without Crossings on Multicolored Points in the Plane. In: Akiyama, J., Ito, H., Sakai, T. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2013. Lecture Notes in Computer Science(), vol 8845. Springer, Cham. https://doi.org/10.1007/978-3-319-13287-7_9

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  • DOI: https://doi.org/10.1007/978-3-319-13287-7_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13286-0

  • Online ISBN: 978-3-319-13287-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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