Coloring hypergraphs defined by stabbed pseudo-disks and abab-free hypergraphs

EYAL ACKERMAN; BALÁZS KESZEGH; DÖMÖTÖR PÁLVÖLGYI

doi:10.1137/19M1290231

Coloring hypergraphs defined by stabbed pseudo-disks and abab-free hypergraphs

EYAL ACKERMAN, BALÁZS KESZEGH, DÖMÖTÖR PÁLVÖLGYI

Faculty of Natural Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

What is the minimum number of colors that always suffice to color every planar set of points such that any disk that contains enough points contains two points of different colors? It is known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely, pseudo-disks, with a nonempty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way, and our proof that they are 3-chromatic is also combinatorial.

Original language	English
Pages (from-to)	2250-2269
Number of pages	20
Journal	SIAM Journal on Discrete Mathematics
Volume	34
Issue number	4
DOIs	https://doi.org/10.1137/19M1290231
State	Published - 2020

Bibliographical note

Funding Information:
\ast Received by the editors September 30, 2019; accepted for publication (in revised form) September 16, 2020; published electronically November 2, 2020. https://doi.org/10.1137/19M1290231 Funding: Research by the first author was partially supported by ERC AdG Disconv and MTA EU10/2016-11001. Research by the second author was supported by the National Research, Development and Innovation Office NKFIH under the grant K 116769. Research by the second and third authors was supported by the Lendület program of the Hungarian Academy of Sciences (MTA) under grant LP2017-19/2017. \dagger Department of Mathematics, Physics, and Computer Science, University of Haifa at Oranim, Tivon 36006, Israel (ackerman@sci.haifa.ac.il). \ddagger Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, and MTA-ELTE Lendület Combinatorial Geometry Research Group, Institute of Mathematics, Eötvös Loránd University (ELTE) Budapest, Hungary (keszegh@gmail.com). \S MTA-ELTE Lendület Combinatorial Geometry Research Group, Institute of Mathematics, Eötvös Loránd University (ELTE), Budapest, Hungary (dom@cs.elte.hu). 1A “region” is merely a set.

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

Keywords

Coloring
Combinatorial geometry
Geometric hypergraph
Pseudoline

ASJC Scopus subject areas

Mathematics (all)

Access to Document

10.1137/19M1290231

Cite this

@article{cea2c05539ca44feba33bcf37835cbd1,

title = "Coloring hypergraphs defined by stabbed pseudo-disks and abab-free hypergraphs",

abstract = "What is the minimum number of colors that always suffice to color every planar set of points such that any disk that contains enough points contains two points of different colors? It is known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely, pseudo-disks, with a nonempty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way, and our proof that they are 3-chromatic is also combinatorial.",

keywords = "Coloring, Combinatorial geometry, Geometric hypergraph, Pseudoline",

author = "EYAL ACKERMAN and BAL{\'A}ZS KESZEGH and D{\"O}M{\"O}T{\"O}R P{\'A}LV{\"O}LGYI",

note = "Funding Information: \ast Received by the editors September 30, 2019; accepted for publication (in revised form) September 16, 2020; published electronically November 2, 2020. https://doi.org/10.1137/19M1290231 Funding: Research by the first author was partially supported by ERC AdG Disconv and MTA EU10/2016-11001. Research by the second author was supported by the National Research, Development and Innovation Office NKFIH under the grant K 116769. Research by the second and third authors was supported by the Lend{\"u}let program of the Hungarian Academy of Sciences (MTA) under grant LP2017-19/2017. \dagger Department of Mathematics, Physics, and Computer Science, University of Haifa at Oranim, Tivon 36006, Israel (ackerman@sci.haifa.ac.il). \ddagger Alfr{\'e}d R{\'e}nyi Institute of Mathematics, Hungarian Academy of Sciences, and MTA-ELTE Lend{\"u}let Combinatorial Geometry Research Group, Institute of Mathematics, E{\"o}tv{\"o}s Lor{\'a}nd University (ELTE) Budapest, Hungary (keszegh@gmail.com). \S MTA-ELTE Lend{\"u}let Combinatorial Geometry Research Group, Institute of Mathematics, E{\"o}tv{\"o}s Lor{\'a}nd University (ELTE), Budapest, Hungary (dom@cs.elte.hu). 1A “region” is merely a set. Publisher Copyright: {\textcopyright} 2020 Society for Industrial and Applied Mathematics.",

year = "2020",

doi = "10.1137/19M1290231",

language = "English",

volume = "34",

pages = "2250--2269",

journal = "SIAM Journal on Discrete Mathematics",

issn = "0895-4801",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "4",

}

TY - JOUR

T1 - Coloring hypergraphs defined by stabbed pseudo-disks and abab-free hypergraphs

AU - ACKERMAN, EYAL

AU - KESZEGH, BALÁZS

AU - PÁLVÖLGYI, DÖMÖTÖR

N1 - Funding Information: \ast Received by the editors September 30, 2019; accepted for publication (in revised form) September 16, 2020; published electronically November 2, 2020. https://doi.org/10.1137/19M1290231 Funding: Research by the first author was partially supported by ERC AdG Disconv and MTA EU10/2016-11001. Research by the second author was supported by the National Research, Development and Innovation Office NKFIH under the grant K 116769. Research by the second and third authors was supported by the Lendület program of the Hungarian Academy of Sciences (MTA) under grant LP2017-19/2017. \dagger Department of Mathematics, Physics, and Computer Science, University of Haifa at Oranim, Tivon 36006, Israel (ackerman@sci.haifa.ac.il). \ddagger Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, and MTA-ELTE Lendület Combinatorial Geometry Research Group, Institute of Mathematics, Eötvös Loránd University (ELTE) Budapest, Hungary (keszegh@gmail.com). \S MTA-ELTE Lendület Combinatorial Geometry Research Group, Institute of Mathematics, Eötvös Loránd University (ELTE), Budapest, Hungary (dom@cs.elte.hu). 1A “region” is merely a set. Publisher Copyright: © 2020 Society for Industrial and Applied Mathematics.

PY - 2020

Y1 - 2020

N2 - What is the minimum number of colors that always suffice to color every planar set of points such that any disk that contains enough points contains two points of different colors? It is known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely, pseudo-disks, with a nonempty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way, and our proof that they are 3-chromatic is also combinatorial.

AB - What is the minimum number of colors that always suffice to color every planar set of points such that any disk that contains enough points contains two points of different colors? It is known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely, pseudo-disks, with a nonempty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way, and our proof that they are 3-chromatic is also combinatorial.

KW - Coloring

KW - Combinatorial geometry

KW - Geometric hypergraph

KW - Pseudoline

UR - http://www.scopus.com/inward/record.url?scp=85096928532&partnerID=8YFLogxK

U2 - 10.1137/19M1290231

DO - 10.1137/19M1290231

M3 - Article

AN - SCOPUS:85096928532

SN - 0895-4801

VL - 34

SP - 2250

EP - 2269

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 4

ER -

Coloring hypergraphs defined by stabbed pseudo-disks and abab-free hypergraphs

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this