 
Summary: Disjoint directed cycles
Noga Alon
Abstract
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.
1 Introduction
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 [6] conjectured
that f(k) = 2k  1 for all k 1. Thomassen [9] 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.
