Cubic graphs and related triangulations on orientable surfaces

Let Sg 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 Sg with 2n vertices is asymptotically cgn5/2(g−1)−1γ2n(2n)!, where γ is an algebraic constant and cg 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 Sg has exactly one non-planar component.