Bipartitions with prescribed order of highly connected digraphs

Abstract: 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 $\delta(D)$ to denote $\mathop{\text{min}}\limits_{v\in V (D)} {|N_D^+(v)\cup N_D^-(v)|}$. In this paper, we show the following result. Let $k, l, n, n_1, n_2 \in \mathbb{N}$ with $n_1+n_2\leq n$ and $n_1,n_2\geq n/20$. Suppose that $D$ is a strongly $10^7k(k+l)^2\log(2kl$)-connected digraph of order $n$ with $\delta(D)\geq n-l$. Then there exist two disjoint subsets $V_1, V_2\in V(D)$ with $|V_1| = n_1$ and $|V_2| = n_2$ such that each of $D[V_1]$, $D[V_2]$, and $D[V_1, V_2]$ is strongly $k$-connected. In particular, $V_1$ and $V_2$ form a partition of $V(D)$ when $n_1+n_2=n$. This result improves the earlier result of Kim, K\"{u}hn, and Osthus [SIAM J. Discrete Math. 30 (2016) 895--911].