Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n   vertices that can be embedded in the orientable surface SgSg of genus g grows asymptotically likec(g)n5(g−1)/2−1γnn!c(g)n5(g−1)/2−1γnn! where c(g)>0c(g)>0, and γ≈27.23γ≈27.23 is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0g=0, obtained by Giménez 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 SgSg has a unique 2-connected component of linear size with high probability.