Colorful Strips

  • Greg Aloupis
  • Jean Cardinal
  • Sébastien Collette
  • Shinji Imahori
  • Matias Korman
  • Stefan Langerman
  • Oded Schwartz
  • Shakhar Smorodinsky
  • Perouz Taslakian
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6034)


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(4ln 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. Lower bounds are also given for all of the above problems. 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Greg Aloupis
    • 1
  • Jean Cardinal
    • 1
  • Sébastien Collette
    • 1
  • Shinji Imahori
    • 2
  • Matias Korman
    • 3
  • Stefan Langerman
    • 1
  • Oded Schwartz
    • 4
  • Shakhar Smorodinsky
    • 5
  • Perouz Taslakian
    • 1
  1. 1.Université Libre de BruxellesBrusselsBelgium
  2. 2.Graduate School of EngineeringNagoya UniversityNagoyaJapan
  3. 3.Graduate School of Information Sciences (GSIS)Tohoku UniversityJapan
  4. 4.Departments of MathematicsTechnische Universität BerlinBerlinGermany
  5. 5.Ben-Gurion UniversityBe’er ShevaIsrael

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