Asymptotic enumeration and limit laws for graphs of fixed genus

摘要

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface Sg of genus g grows asymptotically like. c(g)n5(g-1)/2-1αnn! where c(g)>0, and α≈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. 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 Sg has a unique 2-connected component of linear size with high probability.