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

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

Access to Document

10.1007/s00454-017-9902-y

Cite this

@article{b09ee10de07248a3ac030f6ba76bf1e7,

title = "Coloring Points with Respect to Squares",

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.",

keywords = "Cover-decomposability, Geometric hypergraph coloring, Self-coverability",

author = "Eyal Ackerman and Bal{\'a}zs Keszegh and Mate Vizer",

note = "Publisher Copyright: {\textcopyright} 2017, Springer Science+Business Media New York.",

year = "2017",

month = dec,

day = "1",

doi = "10.1007/s00454-017-9902-y",

language = "English",

volume = "58",

pages = "757--784",

journal = "Discrete and Computational Geometry",

issn = "0179-5376",

publisher = "Springer New York",

number = "4",

}

TY - JOUR

T1 - Coloring Points with Respect to Squares

AU - Ackerman, Eyal

AU - Keszegh, Balázs

AU - Vizer, Mate

PY - 2017/12/1

Y1 - 2017/12/1

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.

AB - 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.

KW - Cover-decomposability

KW - Geometric hypergraph coloring

KW - Self-coverability

UR - http://www.scopus.com/inward/record.url?scp=85021182246&partnerID=8YFLogxK

U2 - 10.1007/s00454-017-9902-y

DO - 10.1007/s00454-017-9902-y

M3 - Article

AN - SCOPUS:85021182246

SN - 0179-5376

VL - 58

SP - 757

EP - 784

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 4

ER -

Coloring Points with Respect to Squares

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this