New bounds on crossing numbers

Janos Pach, Joel Spencer, Geza Toth

Research output: Chapter in Book/Report/Conference proceedingChapter

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 P. Erdos and R. 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 M. Simonovits.

Original languageEnglish (US)
Title of host publicationProceedings of the Annual Symposium on Computational Geometry
PublisherACM
Pages124-133
Number of pages10
StatePublished - 1999
EventProceedings of the 1999 15th Annual Symposium on Computational Geometry - Miami Beach, FL, USA
Duration: Jun 13 1999Jun 16 1999

Other

OtherProceedings of the 1999 15th Annual Symposium on Computational Geometry
CityMiami Beach, FL, USA
Period6/13/996/16/99

ASJC Scopus subject areas

  • Chemical Health and Safety
  • Software
  • Safety, Risk, Reliability and Quality
  • Geometry and Topology

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