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 S g of genus g grows asymptotically like c ( g ) n 5 ( g − 1 ) / 2 − 1 γ n n ! where c ( g ) > 0 , and γ ≈ 27.23 is the exponential growth rate of planar graphs. This generalizes the result for the planar case g = 0 , obtained by Giménez and Noy. logous 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.