Asymptotic enumeration and limit laws for graphs of fixed genus

  • Authors:

  • Guillaume Chapuy;íric Fusy;Omer Giménez;Bojan Mohar;Marc Noy

  • Affiliations:

  • Dept. Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada;Laboratoire d'Informatique de l'ícole Polytechnique, 91128 Palaiseau Cedex, France;Dept. de Llenguatges i Sistemes Informítics, Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain;Dept. Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada;Dept. de Matemítica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2011

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Abstract

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S"g of genus g grows asymptotically likec^(^g^)n^5^(^g^-^1^)^/^2^-^1@c^nn! where c^(^g^)0, and @c~27.23 is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S"g has a unique 2-connected component of linear size with high probability.