Random graphs embeddable in order-dependent surfaces

Given a 'genus function' g = g(n), we let E-g be the class of all graphs G such that if G has order n (i.e., has n vertices) then it is embeddable in a surface of Euler genus at most g(n). Let the random graph R-n be sampled uniformly from the graphs in E-g on vertex set [n] = {1, ... ,n}. Observe that if g(n) is 0 then R-n is a random planar graph, and if g(n) is sufficiently large then R-n is a binomial random graph G(n, 21). We investigate typical properties of R-n. We find that for every genus function g, with high probability at most one component of R-n is non-planar. In contrast, we find a transition for example for connectivity: if g(n) is O(n/ log n) and g is non-decreasing then lim inf(n ->infinity)P(R-n is connected) < 1, and if g(n) >> n then with high probability Rn is connected. These results also hold when we consider orientable and non-orientable surfaces separately. We also investigate random graphs sampled uniformly from the 'hereditary part' or the 'minor-closed part' of E-g, and briefly consider corresponding results for unlabelled graphs.