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 $\epsilon$ with $0<\epsilon<1/2$, there exists an integer $\delta_0$ such that every tournament with minimum out-degree at least $\delta_0$ admits a bisection $(A, B)$, so that each vertex has at least $(1/2-\epsilon)$ of its out-neighbors in $A$, and in $B$ as well.