Let S g be the orientable surface of genus g and denote by  g (n, m) the class of all graphs on vertex set [n] = {1, … , n} with m edges embeddable on S g . We prove that the component structure of a graph chosen uniformly at random from  g (n, m) features two phase transitions. The first phase transition mirrors the classical phase transition in the Erdős-Rényi random graph G(n, m) chosen uniformly at random from all graphs with vertex set [n] and m edges. It takes place at m = n 2 + O(n 2∕3 ), when the giant component emerges. The second phase transition occurs at m = n + O(n 3∕5 ), when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from G(n, m) and has only been observed for graphs on surfaces.

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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 cgn 5(g−1)/2−1 γ 2n (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.

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