Packing arborescences in random digraphs

We study the problem of packing arborescences in the random digraph D(n, p), where each possible arc is included uniformly at random with probability p = p(n). Let λ(D(n, p)) denote the largest integer λ ≥ 0 such that, for all 0 ≤ ` ≤ λ, we have ∑`−1 i=0(`− i)|{v : din(v) = i}| ≤ `. We show that the maximum number of arcdisjoint arborescences in D(n, p) is λ(D(n, p)) a.a.s. We also give tight estimates for λ(D(n, p)) depending on the range of p.