Birth of a giant (k1,k2)-core in the random digraph

The (k1,k2)-core of a digraph is the largest sub-digraph with minimum in-degree and minimum out-degree at least k1 and k2 respectively. For max⁡{k1,k2}≥2, we establish existence of the threshold edge-density c⁎=c⁎(k1,k2), such that the random digraph D(n,m), on the vertex set [n] with m edges, asymptotically almost surely has a giant (k1,k2)-core if m/n>c⁎, and has no (k1,k2)-core if m/n