Summary: Disjoint directed cycles
It is shown that there exists a positive so that for any integer k, every directed graph with
minimum outdegree at least k contains at least k vertex disjoint cycles. On the other hand, for
every k there is a digraph with minimum outdegree k which does not contain two vertex or edge
disjoint cycles of the same length.
All graphs and digraphs considered here contain no parallel edges, unless otherwise specified, but
may have loops. Throughout the paper, a cycle in a directed graph always means a directed cycle.
For a positive integer k, let f(k) denote the smallest integer so that every digraph of minimum
outdegree at least f(k) contains k vertex disjoint cycles. Bermond and Thomassen  conjectured
that f(k) = 2k - 1 for all k 1. Thomassen  showed that this is the case for k 2, and proved
that for every k 2
f(k + 1) (k + 1)(f(k) + k),
thus concluding that f(k) is finite for every k and that f(k) (k+1)!. Here we improve this estimate
and show that f(k) is bounded by a linear function of k.
Theorem 1.1 There exists an absolute constant C so that f(k) Ck for all k. In particular,
C = 64 will do.
An easy corollary of this theorem is the following.