The evolution of random graphs on surfaces

Abstract

For integers g, m ≥ 0 and n > 0, let Sg(n, m) denote the graph taken uniformly at random from the set of all graphs on (1, 2, . . ., n) with exactly m = m(n) edges and with genus at most g. We use counting arguments to investigate the components, subgraphs, maximum degree, and largest face size of Sg(n, m), finding that there is often different asymptotic behavior depending on the ratiomn . In our main results, we show that the probability that Sg(n, m) contains any given nonplanar component converges to 0 as n → ∞ for all m(n), the probability that Sg(n, m) contains a copy of any given planar graph converges to 1 as n → ∞ if lim infmn > 1, the maximum degree of Sg(n, m) is Θ(ln n) with high probability if lim infmn > 1, and the largest face size of Sg(n, m) has a threshold aroundmn = 1, where it changes from Θ(n) to Θ(ln n) with high probability.