Asymptotic Enumeration of Labelled Graphs by Genus

@article{Bender2011AsymptoticEO,
  title={Asymptotic Enumeration of Labelled Graphs by Genus},
  author={Edward A. Bender and Zhicheng Gao},
  journal={Electr. J. Comb.},
  year={2011},
  volume={18}
}
We obtain asymptotic formulas for the number of rooted 2-connected and 3connected 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 facewidth 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 k ≤ 3. 

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