Random graphs embeddable in order-dependent surfaces

Given a ‘genus function’ $g=g(n)$, we let ${\mathcal{E}}^g$ 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 ${\mathcal{E}}^g$ on vertex set $[n]=\{1,\dots ,n\}$. 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 $G(n,\frac{1}{2})$. 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 , and if $g(n)\gg n$ then with high probability $R_n$ 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 ${\mathcal{E}}^g$, and briefly consider corresponding results for unlabelled graphs.