On the largest strong components in m-out digraphs

AbstractA digraph is m-out if every vertex in the digraph has an outdegree equal to m. For n > m ⩾ 2, we consider the m-out digraph D(m, n) obtain by randomly choosing a digraph from the set of all m-out digraphs with n vertices. We show by using a constructive method that for any fixed m ⩾ 2, almost every D(m, n) contains a unique largest strongly connected subgraph and that if N(m, n) is the number of vertices in this subgraph, then n-1N(m, n) converges in probability to (1− y(m)) where y(m) is the smallest root of y = em(y−1)