Random graphs embeddable in order‐dependent surfaces

Abstract

AbstractGiven a ‘genus function’ , we let be the class of all graphs such that if has order (i.e., has vertices) then it is embeddable in a surface of Euler genus at most . Let the random graph be sampled uniformly from the graphs in on vertex set . Observe that if is 0 then is a random planar graph, and if is sufficiently large then is a binomial random graph . We investigate typical properties of . We find that for every genus function , with high probability at most one component of is non‐planar. In contrast, we find a transition for example for connectivity: if is and is non‐decreasing then , and if then with high probability 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 , and briefly consider corresponding results for unlabelled graphs.