Abstract. Let [Formula: see text] be a digraph of order [Formula: see text]. We define the degree of vertex [Formula: see text] in [Formula: see text] to be [Formula: see text], where [Formula: see text] and [Formula: see text] are the out-degree and in-degree of [Formula: see text] in [Formula: see text], respectively. Let [Formula: see text] be a positive integer and let [Formula: see text] be any given subset of [Formula: see text] with [Formula: see text]. In this paper we show that if [Formula: see text] for all [Formula: see text], then for any integer partition [Formula: see text] with [Formula: see text] for each [Formula: see text], there are [Formula: see text] disjoint cycles containing exactly [Formula: see text] vertices of [Formula: see text], respectively. The degree condition [Formula: see text] is sharp in some sense and this result confirms the conjecture posed by Wang [J. Graph Theory, 34 (2000), pp. 154–162] as a corollary. The result in this paper implies a theorem on cycle-factors containing matchings in bipartite graphs. Further, the special case [Formula: see text] is a directed version of the Aigner–Brandt theorem on disjoint cycles in graphs.

We conjectured in [H. Wang, Australas. J. Combin., 19 (1999), pp. 115--121] that, for each integer $k\geq 2$, there exists N(k) such that if G=(V 1 ,V 2 ;E) is a bipartite graph with $|V_1|=|V_2|=n\geq N(k)$ and d(x)+d(y)\geq n+k$ for each pair of nonadjacent vertices x and y of G with $x\in V_1$ and $y\in V_2$, then for any k independent edges e 1 , . . ., e k of G, there exist k vertex-disjoint cycles C 1 , . . . ,C k in G such that $e_i\in E(C_i)$ for all $i\in\{1,\ldots ,k\}$ and $V(C_1\cup\cdots \cup C_k)=V(G)$. This conjecture is also verified for k=2 in [H. Wang, Australas. J. Combin., 19 (1999), pp. 115--121]. We prove this conjecture for k=3 in this paper.