How Many Ways Can One Draw A Graph?

Using results from extremal graph theory, we determine the asymptotic number of string graphs with n vertices, i.e., graphs that can be obtained as the intersection graph of a system of continuous arcs in the plane. The number becomes much smaller, for any fixed d, if we restrict our attention to systems of arcs, any two of which cross at most d times. As an application, we estimate the number of different drawings of the complete graph K n with n vertices under various side conditions.

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Correspondence to János Pach*.

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Dedicated to Miklós Simonovits on his sixtieth birthday

* Supported by NSF grant CR-00-98246, PSC-CUNY Research Award 62450-0031 and OTKA-T-032452.

† Supported by OTKA-T-032452 and OTKA-T-038397.

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Pach*, J., Tóth†, G. How Many Ways Can One Draw A Graph?. Combinatorica 26, 559–576 (2006).

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Mathematics Subject Classification (2000):

  • 05A16
  • 05C30
  • 05C62