Cubic graphs and related triangulations on orientable surfaces
Résumé
Let Sg be the orientable surface of genus g. We show that the number of vertex-labelled cubic multigraphs embeddable on Sg with 2n vertices is asymptotically cgn5(g−1)/2−1γ2n(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 we prove that a typical cubic multigraph embeddable on Sg, g≥1, has exactly one non-planar component.