Cubic graphs and related triangulations on orientable surfaces

TL;DR

This paper derives asymptotic formulas for counting vertex-labelled cubic multigraphs embeddable on orientable surfaces of genus g, revealing structural properties of typical such graphs, including the prevalence of a single non-planar component.

Contribution

It provides the first asymptotic enumeration formulas for cubic multigraphs on orientable surfaces of arbitrary genus, including simple and weighted variants, and analyzes their typical structure.

Findings

Asymptotic count of cubic multigraphs on surfaces of genus g

Extension to simple and weighted cubic graphs

Most such graphs have exactly one non-planar component

Abstract

Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We show that the number of vertex-labelled cubic multigraphs embeddable on $\mathbb{S}_g$ with $2n$ vertices is asymptotically $c_g n^{5(g-1)/2-1}\gamma^{2n}(2n)!$, 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 we prove that a typical cubic multigraph embeddable on $\mathbb{S}_g$, $g\ge 1$, has exactly one non-planar component.