Cubic Graphs and Related Triangulations on Orientable Surfaces

Abstract

Let $\mathbb{S}_g$ 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 $\mathbb{S}_g$ 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 $g\ge1$, we prove that a typical cubic multigraph embeddable on $\mathbb{S}_g$ has exactly one non-planar component.