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Asymptotic enumeration of labelled graphs by genus. (English) Zbl 1205.05013

Summary: 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 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 \(k \leq 3\).

MSC:

05A16 Asymptotic enumeration
05C30 Enumeration in graph theory
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
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