Partition and Disjoint Cycles in Digraphs

Let D be a digraph, we use δ + ( 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 δ + ( D ) ≥ F ( d 1 , … , d k ) , then V ( D ) can be partitioned into k parts V 1 , … , V k with δ + ( 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 Lovász 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.