Cubic Graphs and Related Triangulations on Orientable Surfaces.

Let S-g be the orientable surface of genus g for a fixed non-negative integer g. We show that the number of vertex-labelled cubic multigraphs embeddable on S-g with 2n vertices is asymptotically C(g)n(5/2( g -1)-1)gamma(2n)(2n)!, where gamma is an algebraic constant and C-g is a constant depending only on the genus g. We also derive an analogous result for simple cubic graphs and weighted cubic multigraphs. Additionally, for g >= 1, we prove that a typical cubic multigraph embeddable on S-g has exactly one non-planar component.