Coloring points with respect to squares

Eyal Ackerman; Balázs Keszegh; Máté Vizer

doi:10.4230/LIPIcs.SoCG.2016.5

Coloring points with respect to squares

Eyal Ackerman, Balázs Keszegh, Máté Vizer

Faculty of Natural Sciences

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › 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 S of points in the plane 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 homothets of a fixed parallelogram.

Original language	English
Title of host publication	32nd International Symposium on Computational Geometry, SoCG 2016
Editors	Sandor Fekete, Anna Lubiw
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages	5.1-5.16
ISBN (Electronic)	9783959770095
DOIs	https://doi.org/10.4230/LIPIcs.SoCG.2016.5
State	Published - 1 Jun 2016
Event	32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States Duration: 14 Jun 2016 → 17 Jun 2016

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	51
ISSN (Print)	1868-8969

Conference

Conference	32nd International Symposium on Computational Geometry, SoCG 2016
Country/Territory	United States
City	Boston
Period	14/06/16 → 17/06/16

Bibliographical note

Publisher Copyright:
© Eyal Ackerman, Balázs Keszegh, and Máté Vizer.

Keywords

Cover-decomposability
Geometric hypergraph coloring
Homothets
Polychromatic coloring

ASJC Scopus subject areas

Software

Access to Document

10.4230/LIPIcs.SoCG.2016.5

Cite this

Ackerman, E., Keszegh, B., & Vizer, M. (2016). Coloring points with respect to squares. In S. Fekete, & A. Lubiw (Eds.), 32nd International Symposium on Computational Geometry, SoCG 2016 (pp. 5.1-5.16). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 51). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2016.5

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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 S of points in the plane 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 homothets of a fixed parallelogram.",

keywords = "Cover-decomposability, Geometric hypergraph coloring, Homothets, Polychromatic coloring",

author = "Eyal Ackerman and Bal{\'a}zs Keszegh and M{\'a}t{\'e} Vizer",

note = "Publisher Copyright: {\textcopyright} Eyal Ackerman, Bal{\'a}zs Keszegh, and M{\'a}t{\'e} Vizer.; 32nd International Symposium on Computational Geometry, SoCG 2016 ; Conference date: 14-06-2016 Through 17-06-2016",

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series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

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Ackerman, E, Keszegh, B & Vizer, M 2016, Coloring points with respect to squares. in S Fekete & A Lubiw (eds), 32nd International Symposium on Computational Geometry, SoCG 2016. Leibniz International Proceedings in Informatics, LIPIcs, vol. 51, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 5.1-5.16, 32nd International Symposium on Computational Geometry, SoCG 2016, Boston, United States, 14/06/16. https://doi.org/10.4230/LIPIcs.SoCG.2016.5

Coloring points with respect to squares. / Ackerman, Eyal; Keszegh, Balázs; Vizer, Máté.
32nd International Symposium on Computational Geometry, SoCG 2016. ed. / Sandor Fekete; Anna Lubiw. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. p. 5.1-5.16 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 51).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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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 S of points in the plane 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 homothets of a fixed parallelogram.

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KW - Geometric hypergraph coloring

KW - Homothets

KW - Polychromatic coloring

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Coloring points with respect to squares

Abstract

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Keywords

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