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Colorful Strips

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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.

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Authors and Affiliations

  1. Université Libre de Bruxelles, Brussels, Belgium

    Greg Aloupis, Jean Cardinal, Sébastien Collette, Matias Korman, Stefan Langerman & Perouz Taslakian

  2. Nagoya University, Nagoya, Japan

    Shinji Imahori

  3. The Weizmann Institute of Science, Rehovot, Israel

    Oded Schwartz

  4. Ben-Gurion University, Be’er Sheva, Israel

    Shakhar Smorodinsky

Authors
  1. Greg Aloupis
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  2. Jean Cardinal
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  3. Sébastien Collette
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  4. Shinji Imahori
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  5. Matias Korman
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  6. Stefan Langerman
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  7. Oded Schwartz
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  8. Shakhar Smorodinsky
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  9. Perouz Taslakian
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Corresponding author

Correspondence to Greg Aloupis.

Additional information

Chargé de Recherches du FRS-FNRS.

Maître de Recherches du FRS-FNRS.

Supported by the Communauté française de Belgique - ARC.

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Aloupis, G., Cardinal, J., Collette, S. et al. Colorful Strips. Graphs and Combinatorics 27, 327–339 (2011). https://doi.org/10.1007/s00373-011-1014-5

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  • DOI: https://doi.org/10.1007/s00373-011-1014-5

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