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 behaviour depending on the ratio m n . In our main results, we show that the probability that Sg(n, m) contains any given non-planar 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 inf m n > 1; the maximum degree of Sg(n, m) is Θ(ln n) with high probability if lim inf m n > 1; and the largest face size of Sg(n, m) has a threshold around m n = 1 where it changes from Θ(n) to Θ(ln n) with high probability.

We investigate the genus g(n,m) of the Erdős‐Rényi random graph G(n,m), providing a thorough description of how this relates to the function m = m(n), and finding that there is different behavior depending on which “region” m falls into. Results already exist for m≤n2+Ofalse(n2false/3false) and m=ω()n1+1j for j∈double-struckN, and so we focus on the intermediate cases. We establish that gfalse(n,mfalse)=false(1+ofalse(1false)false)m2 whp (with high probability) when n ≪ m = n1 + o(1), that g(n,m) = (1 + o(1))μ(λ)m whp for a given function μ(λ) when m∼λn for λ>12, and that gfalse(n,mfalse)=false(1+ofalse(1false)false)8s33n2 whp when m=n2+s for n2/3 ≪ s ≪ n. We then also show that the genus of a fixed graph can increase dramatically if a small number of random edges are added. Given any connected graph with bounded maximum degree, we find that the addition of ϵn edges will whp result in a graph with genus Ω(n), even when ϵ is an arbitrarily small constant! We thus call this the “fragile genus” property.

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 behaviour depending on the ratio m n . In our main results, we show that the probability that Sg(n, m) contains any given non-planar 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 inf m n > 1; the maximum degree of Sg(n, m) is Θ(ln n) with high probability if lim inf m n > 1; and the largest face size of Sg(n, m) has a threshold around m n = 1 where it changes from Θ(n) to Θ(ln n) with high probability.