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arXiv · 1603.01440

Cubic graphs and related triangulations on orientable surfaces

Wenjie Fang·Mihyun Kang·Michael Moßhammer·Philipp Sprüssel
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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.

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Wenjie Fang, Mihyun Kang, Michael Moßhammer, Philipp Sprüssel. 2016-03-04. Cubic graphs and related triangulations on orientable surfaces. https://arxiv.org/abs/1603.01440

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