On splitting digraphs

Abstract

In 1995, Stiebitz 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, multipartite tournaments, and digraphs with bounded maximum in-degrees. In particular, we show that for every ϵ with 0<ϵ<1∕2, 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 (1∕2−ϵ) of its out-neighbors in A, and in B as well.