### Abstract

The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by κ(n, e) the minimum of cr(G) taken over all graphs with n vertices and at least e edges. We prove a conjecture of Erdos and Guy by showing that κ(n, e)n^{2}/e^{3} tends to a positive constant as n → ∞ and n ≪ e ≪ n^{2}. Similar results hold for graph drawings on any other surface of fixed genus. We prove better bounds for graphs satisfying some monotone properties. In particular, we show that if G is a graph with n vertices and e ≥ 4n edges, which does not contain a cycle of length four (resp. six), then its crossing number is at least ce^{4}/n^{3} (resp. ce^{5}/n^{4}), where c > 0 is a suitable constant. These results cannot be improved, apart from the value of the constant. This settles a question of Simonovits.

Original language | English (US) |
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Pages (from-to) | 623-644 |

Number of pages | 22 |

Journal | Discrete and Computational Geometry |

Volume | 24 |

Issue number | 4 |

State | Published - 2000 |

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### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

*24*(4), 623-644.

**New Bounds on Crossing Numbers.** / Pach, J.; Spencer, J.; Tóth, G.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 24, no. 4, pp. 623-644.

}

TY - JOUR

T1 - New Bounds on Crossing Numbers

AU - Pach, J.

AU - Spencer, J.

AU - Tóth, G.

PY - 2000

Y1 - 2000

N2 - The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by κ(n, e) the minimum of cr(G) taken over all graphs with n vertices and at least e edges. We prove a conjecture of Erdos and Guy by showing that κ(n, e)n2/e3 tends to a positive constant as n → ∞ and n ≪ e ≪ n2. Similar results hold for graph drawings on any other surface of fixed genus. We prove better bounds for graphs satisfying some monotone properties. In particular, we show that if G is a graph with n vertices and e ≥ 4n edges, which does not contain a cycle of length four (resp. six), then its crossing number is at least ce4/n3 (resp. ce5/n4), where c > 0 is a suitable constant. These results cannot be improved, apart from the value of the constant. This settles a question of Simonovits.

AB - The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by κ(n, e) the minimum of cr(G) taken over all graphs with n vertices and at least e edges. We prove a conjecture of Erdos and Guy by showing that κ(n, e)n2/e3 tends to a positive constant as n → ∞ and n ≪ e ≪ n2. Similar results hold for graph drawings on any other surface of fixed genus. We prove better bounds for graphs satisfying some monotone properties. In particular, we show that if G is a graph with n vertices and e ≥ 4n edges, which does not contain a cycle of length four (resp. six), then its crossing number is at least ce4/n3 (resp. ce5/n4), where c > 0 is a suitable constant. These results cannot be improved, apart from the value of the constant. This settles a question of Simonovits.

UR - http://www.scopus.com/inward/record.url?scp=0034369415&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034369415&partnerID=8YFLogxK

M3 - Article

VL - 24

SP - 623

EP - 644

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 4

ER -