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Discrete & Computational Geometry

, Volume 24, Issue 4, pp 623–644 | Cite as

New Bounds on Crossing Numbers

  • J. Pach
  • J. Spencer
  • G. Tóth

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 os and Guy by showing that κ(n,e)n 2 /e 3 tends to a positive constant as n→∈fty and n l e l 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.

Keywords

Positive Constant Monotone Property Fixed Genus Good Bound Suitable Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag New York Inc. 2000

Authors and Affiliations

  • J. Pach
    • 1
  • J. Spencer
    • 1
  • G. Tóth
    • 2
  1. 1.Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA pach@cims.nyu.edu spencer@cs.nyu.edu US
  2. 2.Mathematical Institute, Hungarian Academy of Sciences, PF 127, H-1364 Budapest, Hungary HU

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