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 (D) to denote minₕ ₕ (₃) |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) ² (2kl) -connected digraph of order n with (D) n-l. Then there exist two disjoint subsets V₁, V₂ V (D) with |V₁| = n₁ and |V₂| = n₂ such that each of DV₁, DV₂, and DV₁, 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\"uhn, and Osthus SIAM J. Discrete Math. 30 (2016) 895--911.