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Graphs and Combinatorics

, Volume 27, Issue 3, pp 327–339 | Cite as

Colorful Strips

  • Greg Aloupis
  • Jean Cardinal
  • Sébastien Collette
  • Shinji Imahori
  • Matias Korman
  • Stefan Langerman
  • Oded Schwartz
  • Shakhar Smorodinsky
  • Perouz Taslakian
Proceedings Paper

Abstract

We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k − 1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k − 1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.

Keywords

Hypergraph coloring Covering decomposition Lovász local lemma Computational geometry 

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Copyright information

© Springer 2011

Authors and Affiliations

  • Greg Aloupis
    • 1
  • Jean Cardinal
    • 1
  • Sébastien Collette
    • 1
  • Shinji Imahori
    • 2
  • Matias Korman
    • 1
  • Stefan Langerman
    • 1
  • Oded Schwartz
    • 3
  • Shakhar Smorodinsky
    • 4
  • Perouz Taslakian
    • 1
  1. 1.Université Libre de BruxellesBrusselsBelgium
  2. 2.Nagoya UniversityNagoyaJapan
  3. 3.The Weizmann Institute of ScienceRehovotIsrael
  4. 4.Ben-Gurion UniversityBe’er ShevaIsrael

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