New Bounds on Crossing Numbers

J. Pach, J. Spencer, G. Tóth

Research output: Contribution to journalArticle

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)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.

Original languageEnglish (US)
Pages (from-to)623-644
Number of pages22
JournalDiscrete and Computational Geometry
Volume24
Issue number4
StatePublished - 2000

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Crossing number
Graph in graph theory
Graph Drawing
Erdös
Monotone
Genus
Tend
Denote
Cycle

ASJC Scopus subject areas

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

Cite this

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

In: Discrete and Computational Geometry, Vol. 24, No. 4, 2000, p. 623-644.

Research output: Contribution to journalArticle

Pach, J, Spencer, J & Tóth, G 2000, 'New Bounds on Crossing Numbers', Discrete and Computational Geometry, vol. 24, no. 4, pp. 623-644.
Pach, J. ; Spencer, J. ; Tóth, G. / New Bounds on Crossing Numbers. In: Discrete and Computational Geometry. 2000 ; Vol. 24, No. 4. pp. 623-644.
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