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

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} be a set of multicolored points in the plane suc...

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Bibliographic Details
Published inDiscrete and Computational Geometry and Graphs Vol. 8845; pp. 96 - 111
Main Authors Kano, Mikio, Suzuki, Kazuhiro, Uno, Miyuki
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 01.01.2014
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
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Summary:Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} be a set of multicolored points in the plane such that no three points are collinear and each color appears on at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lceil |X|/2 \rceil $$\end{document} points. We show the existence of a non-crossing properly colored geometric perfect matching on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} (if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|X|$$\end{document} is even), and the existence of a non-crossing properly colored geometric spanning tree with maximum degree at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document}. 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.
Bibliography: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.
ISBN:9783319132860
3319132865
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-319-13287-7_9