Polychromatic coloring for half-planes

Shakhar Smorodinsky, Yelena Yuditsky

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

We prove that for every integer k, every finite set of points in the plane can be k-colored so that every half-plane that contains at least 2k-1 points, also contains at least one point from every color class. We also show that the bound 2k-1 is best possible. This improves the best previously known lower and upper bounds of 43k and 4k-1 respectively. We also show that every finite set of half-planes can be k-colored so that if a point p belongs to a subset Hp of at least 3k-2 of the half-planes then Hp contains a half-plane from every color class. This improves the best previously known upper bound of 8k-3. Another corollary of our first result is a new proof of the existence of small size ε-nets for points in the plane with respect to half-planes.

Original languageEnglish
Pages (from-to)146-154
Number of pages9
JournalJournal of Combinatorial Theory - Series A
Volume119
Issue number1
DOIs
StatePublished - 1 Jan 2012

Keywords

  • Cover decomposable
  • Discrete geometry
  • Epsilon nets
  • Polychromatic coloring

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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