Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

  • Andrei Asinowski
  • Jean Cardinal
  • Nathann Cohen
  • Sébastien Collette
  • Thomas Hackl
  • Michael Hoffmann
  • Kolja Knauer
  • Stefan Langerman
  • Michał Lasoń
  • Piotr Micek
  • Günter Rote
  • Torsten Ueckerdt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)

Abstract

We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color.

We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k) = 3k − 2. This can be interpreted as coloring point sets in ℝ2 with k colors such that any bottomless rectangle containing at least 3k − 2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set. For this problem, we also prove a lower bound p(k) > ck, where c > 1.67. Hence, for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors.

Chen et al. (2009) proved that no such function p(k) exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Pálvölgyi on cover-decomposability of octants (2011, 2012).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Andrei Asinowski
    • 1
  • Jean Cardinal
    • 2
  • Nathann Cohen
    • 3
  • Sébastien Collette
    • 2
  • Thomas Hackl
    • 4
  • Michael Hoffmann
    • 5
  • Kolja Knauer
    • 6
  • Stefan Langerman
    • 2
  • Michał Lasoń
    • 7
  • Piotr Micek
    • 7
  • Günter Rote
    • 1
  • Torsten Ueckerdt
    • 8
  1. 1.Freie Universität BerlinGermany
  2. 2.Université Libre de BruxellesBelgium
  3. 3.Université Paris-Sud 11France
  4. 4.TU GrazAustria
  5. 5.ETH ZürichSwitzerland
  6. 6.Université Montpellier 2France
  7. 7.Jagiellonian University in KrakowPoland
  8. 8.Karlsruhe Institute of TechnologyGermany

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