Asymptotic enumeration and limit laws for graphs of fixed genus

Details Asymptotic enumeration and limit laws for graphs of fixed genus Asymptotic enumeration and limit laws for graphs of fixed genus

2010-01-20

arxiv.org/abs/1001.3628v1

Journal of Combinatorial Theory, Series A, 118(3):748-777 (2011)

Mathematics

Combinatorics

Scientific paper

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 like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ 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.

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