On the Degenerate Crossing Number

Eyal Ackerman; Rom Pinchasi

doi:10.1007/s00454-013-9493-1

On the Degenerate Crossing Number

Eyal Ackerman, Rom Pinchasi

Faculty of Natural Sciences

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

Abstract

The degenerate crossing number cr^*(G) of a graph G is the minimum number of crossing points of edges in any drawing of G as a simple topological graph in the plane. This notion was introduced by Pach and Tóth who showed that for a graph G with n vertices and e ≥ 4n edges cr^*(G)=Ω(e⁴ / n⁴). In this paper we completely resolve the main open question about degenerate crossing numbers and show that cr*(G)=Ω (e³ / n²), provided that e≥ 4n. This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.

Original language	English
Pages (from-to)	695-702
Number of pages	8
Journal	Discrete and Computational Geometry
Volume	49
Issue number	3
DOIs	https://doi.org/10.1007/s00454-013-9493-1
State	Published - Apr 2013

Bibliographical note

Funding Information:
Rom Pinchasi was supported by ISF Grant (Grant No. 1357/12) and by BSF Grant (Grant No. 2008290).

Keywords

Crossing Lemma
Crossing number
Simple topological graphs

ASJC Scopus subject areas

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

Access to Document

10.1007/s00454-013-9493-1

Cite this

@article{5a9736b88e244f398ebb69b5ae267cb6,

title = "On the Degenerate Crossing Number",

abstract = "The degenerate crossing number cr*(G) of a graph G is the minimum number of crossing points of edges in any drawing of G as a simple topological graph in the plane. This notion was introduced by Pach and T{\'o}th who showed that for a graph G with n vertices and e ≥ 4n edges cr*(G)=Ω(e4 / n4). In this paper we completely resolve the main open question about degenerate crossing numbers and show that cr*(G)=Ω (e3 / n2), provided that e≥ 4n. This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.",

keywords = "Crossing Lemma, Crossing number, Simple topological graphs",

author = "Eyal Ackerman and Rom Pinchasi",

note = "Funding Information: Rom Pinchasi was supported by ISF Grant (Grant No. 1357/12) and by BSF Grant (Grant No. 2008290).",

year = "2013",

month = apr,

doi = "10.1007/s00454-013-9493-1",

language = "English",

volume = "49",

pages = "695--702",

journal = "Discrete and Computational Geometry",

issn = "0179-5376",

publisher = "Springer New York",

number = "3",

}

TY - JOUR

T1 - On the Degenerate Crossing Number

AU - Ackerman, Eyal

AU - Pinchasi, Rom

N1 - Funding Information: Rom Pinchasi was supported by ISF Grant (Grant No. 1357/12) and by BSF Grant (Grant No. 2008290).

PY - 2013/4

Y1 - 2013/4

N2 - The degenerate crossing number cr*(G) of a graph G is the minimum number of crossing points of edges in any drawing of G as a simple topological graph in the plane. This notion was introduced by Pach and Tóth who showed that for a graph G with n vertices and e ≥ 4n edges cr*(G)=Ω(e4 / n4). In this paper we completely resolve the main open question about degenerate crossing numbers and show that cr*(G)=Ω (e3 / n2), provided that e≥ 4n. This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.

AB - The degenerate crossing number cr*(G) of a graph G is the minimum number of crossing points of edges in any drawing of G as a simple topological graph in the plane. This notion was introduced by Pach and Tóth who showed that for a graph G with n vertices and e ≥ 4n edges cr*(G)=Ω(e4 / n4). In this paper we completely resolve the main open question about degenerate crossing numbers and show that cr*(G)=Ω (e3 / n2), provided that e≥ 4n. This bound is best possible (apart for the multiplicative constant) as it matches the tight lower bound for the standard crossing number of a graph.

KW - Crossing Lemma

KW - Crossing number

KW - Simple topological graphs

UR - http://www.scopus.com/inward/record.url?scp=84876451932&partnerID=8YFLogxK

U2 - 10.1007/s00454-013-9493-1

DO - 10.1007/s00454-013-9493-1

M3 - Article

AN - SCOPUS:84876451932

SN - 0179-5376

VL - 49

SP - 695

EP - 702

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 3

ER -

On the Degenerate Crossing Number

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this