Counting strongly connected (k1, k2)-directed cores

Consider the set of all digraphs on [N] with M edges, whose minimum in-degree and minimum out-degree are at least k(1) and k(2) respectively. For k:=min?{k1,k2}2 and M/Nmax?{k1,k2}+,M=(N), we show that, among those digraphs, the fraction of k-strongly connected digraphs is 1-O(N-(k-1)). Earlier with Dan Poole we identified a sharp edge-density threshold c(k1,k2) for birth of a giant (k(1), k(2))-core in the random digraph D(n,m=[cn]). Combining the claims, for c>c(k1,k2) with probability 1-O(N-(k-1)) the giant (k(1), k(2))-core exists and is k-strongly connected.