Bipartitions with prescribed order of highly connected digraphs

A digraph is strongly connected if it has a directed path from x to y for every ordered pair of distinct vertices $x, y$ and it is strongly k-connected if it has at least $k+1$ vertices and remains strongly connected when we delete any set of at most $k-1$ vertices. For a digraph D, we use δ(D) to denote minv∈ V (D) |ND⁺(v)∪ ND⁻(v)|. In this paper, we show the following result. Let k, l, n, n₁, n₂ ∈ N with n₁+n₂≤ n and n₁,n₂≥ n/20. Suppose that D is a strongly 10⁷k(k+l)²log(2kl)-connected digraph of order n with δ(D)≥ n-l. Then there exist two disjoint subsets V₁, V₂∈ V(D) with |V₁| = n₁ and |V₂| = n₂ such that each of D[V₁], D[V₂], and D[V₁, V₂] is strongly k-connected. In particular, V₁ and V₂ form a partition of $V(D)$ when n₁+n₂=n. This result improves the earlier result of Kim, K\"{u}hn, and Osthus [SIAM J. Discrete Math. 30 (2016) 895--911].