Asymptotic Enumeration of Labelled Graphs by Genus - Bender, Edward A.; Gao, Zhicheng (The Electronic Journal of Combinatorics, 2011)
We obtain asymptotic formulas for the number of rooted 2-connected and 3-connected surface maps on an orientable surface of genus g with respect to vertices and edges simultaneously. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive asymptotic formulas for the number of labelled k-connected graphs of orientable genus g for k3.