A note on digraph splitting

A tantalizing open problem, posed independently by Stiebitz in 1995 and by Alon in 1996 and again in 2006, asks whether for every pair of integers s,t≥1 there exists a finite number F(s,t) such that the vertex set of every digraph of minimum out-degree at least F(s,t) can be partitioned into non-empty parts A and B such that the subdigraphs induced on A and B have minimum out-degree at least s and t , respectively. In this short note, we prove that if F(2,2) exists, then all the numbers F(s,t) with s,t≥1 exist and satisfy F(s,t)=Θ(s+t) . In consequence, the problem of Alon and Stiebitz reduces to the case s=t=2 . Moreover, the numbers F(s,t) with s,t≥2 either all exist and grow linearly, or all of them do not exist.