DISJOINT CYCLES IN A DIGRAPH WITH PARTIAL DEGREE.

Let D=(V,A) be a digraph of order n. We define the degree of vertex v in D to be dD(v)=d+D(v)+d-D(v), where d+D(v) and d-D(v) are the out-degree and in-degree of v in D, respectively. Let k be a positive integer and let W be any given subset of V with |W|≥2k. In this paper we show that if dD(x)+dD(y)≥3n-3 for all {x,y}⊆W, then for any integer partition |W|=∑ki=1ni with ni≥2 for each i, there are k disjoint cycles containing exactly n1,...,nk vertices of W, respectively. The degree condition dD(x)+dD(y)≥3n-3 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 W=V is a directed version of the Aigner-Brandt theorem on disjoint cycles in graphs. [ABSTRACT FROM AUTHOR]