Partition and Disjoint Cycles in Digraphs

Let D be a digraph, we use d(+) (D) to denote the minimum out-degree of D. In 2006, Alon proposed a problem stating that if there exists an integer function F(d(1), . . . , d(k)) for a digraph D such that if d(+) (D) = F(d(1), ... , d(k)), then V(D) can be partitioned into k parts V-1, ... , V-k with d(+) (D[V-i]) = d(i) for each i ? [k], here D[V-i] denotes the induced subdigraph of V-i .We prove that F(d(1), ... , d(k)) = 2(d(1) + . . . + d(k)) under the condition that the maximum in-degree is bounded and ln k/2 < min{d(1), ... , d(k)} by using Lovasz Local Lemma. Furthermore, we show that some regular digraphs, and digraphs of small order can be partitioned into k parts such that both the minimum in-degree and the minimum out-degree of the digraph induced by each part are at least d(i) for each i ? [k]. Based on the results above, we further give lower bounds of the minimum out-degree of some special class digraphs containing k vertex disjoint cycles of different lengths.