Coloring Points with Respect to Squares

Eyal Ackerman, Balázs Keszegh, Mate Vizer

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set of points in the plane S⊂ R2 can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering 2-coloring points with respect to homothets of a fixed parallelogram.

Original languageEnglish
Pages (from-to)757-784
Number of pages28
JournalDiscrete and Computational Geometry
Volume58
Issue number4
DOIs
StatePublished - 1 Dec 2017

Bibliographical note

Funding Information:
E. Ackerman’s research partially supported by ERC Advanced Research Grant No. 267165 (DISCONV). B. Keszegh’s research supported by the National Research, Development and Innovation Office – NKFIH under the Grant PD 108406 and K 116769 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. M. Vizer’s research supported by the National Research, Development and Innovation Office – NKFIH under the Grant SNN 116095. We thank the reviewers for reading our paper and for their valuable remarks. In particular, suggestions of one of reviewers helped to simplify some basic lemmas and improve the constant in our main theorem.

Publisher Copyright:
© 2017, Springer Science+Business Media New York.

Keywords

  • Cover-decomposability
  • Geometric hypergraph coloring
  • Self-coverability

ASJC Scopus subject areas

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

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