On the Genus of a Random Graph

Let p = p(n) be a function of n with 0 < p < 1. We consider the random graph model G(n, p); that is, the probability space of simple graphs with vertex–set {1, 2, . . . , n} where two distinct vertices are adjacent with probability p and for distinct pairs these events are mutually independent. Archdeacon and Grable have shown that if p2(1−p2) ≥ 8(logn)/n, then the (orientable) genus of a random graph in G(n, p) is (1+o(1))pn/12. We prove that for every integer i ≥ 1, if n p n, then the genus of a random graph in G(n, p) is (1+o(1)) i 4(i+2) pn. If p = cn, where c is a constant, then the genus of a random graph in G(n, p) is (1+o(1))g(i, c, n)pn for some function g(i, c, n) with 1 12 ≤ g(i, c, n) ≤ 1, but for i > 1 we were unable to compute this function.