Disjoint directed cycles
It is shown that there exists a positive \(\varepsilon\) so that for any integer \(k\), every directed graph with minimum outdegree at least \(k\) contains at least \(\varepsilon k\) 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.
scientific article; zbMATH DE number 954821