Coloring Points with Respect to Squares

Eyal Ackerman; Balázs Keszegh; Mate Vizer

doi:10.1007/s00454-017-9902-y

Coloring Points with Respect to Squares

Eyal Ackerman, Balázs Keszegh, Mate Vizer

Faculty of Natural Sciences

Research output: Contribution to journal › Article › peer-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 language	English
Pages (from-to)	757-784
Number of pages	28
Journal	Discrete and Computational Geometry
Volume	58
Issue number	4
DOIs	https://doi.org/10.1007/s00454-017-9902-y
State	Published - 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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10.1007/s00454-017-9902-y

Cite this

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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.",

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N2 - 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.

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Coloring Points with Respect to Squares

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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