On splitting digraphs

In 1995, Stiebitz [13] asked the following question: For any positive integers s, t, is there a finite integer f(s, t) such that every digraph D with minimum out-degree at least f(s, t) admits a bipartition (A,B) such that A induces a subdigraph with minimum out-degree at least s and B induces a subdigraph with minimum out-degree at least t? We give an affirmative answer for tournaments, bipartite tournaments, multipartite tournaments, and digraphs with bounded maximum in-degrees. In particular, we show that for every 0 < ǫ < 1 4 , there exists an integer δ0 such that every tournament with minimum out-degree at least δ0 admits a bisection (A,B), so that each vertex has at least (14 − ǫ) of its out-neighbors in A, and in B as well.