Partition and Disjoint Cycles in Digraphs

Abstract

Let D be a digraph, we use $\delta ^+(D)$ to denote the minimum out-degree of D. In 2006, Alon proposed a problem stating that if there exists an integer function $F(d_1, \ldots ,d_k)$ for a digraph D such that if $\delta ^{+}(D) \ge F(d_1, \ldots ,d_k)$, then V(D) can be partitioned into k parts $V_1,\ldots ,V_k$ with $\delta ^{+}(D[V_i]) \ge d_i$ for each $i \in [k]$, here $D[V_i]$ denotes the induced subdigraph of $V_i$. We prove that $F(d_1, \ldots ,d_k) \le 2(d_1+\cdots +d_k)$ under the condition that the maximum in-degree is bounded and $\frac{\ln k}{2} < \min \{d_1, \dots , d_k\}$ 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 $i \in [k]$. 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.