A step toward the Bermond-Thomassen conjecture about disjoint cycles in digraphs

International audienceIn 1981, Bermond and Thomassen conjectured that every digraph with minimum out-degree at least 2k-1 contains k disjoint cycles. This conjecture is trivial for k=1, and was established for k=2 by Thomassen in 1983. We verify it for the next case, proving that every digraph with minimum out-degree at least five contains three disjoint cycles. To show this, we improve Thomassen's result by proving that every digraph whose vertices have out-degree at least three, except at most two with out-degree two, indeed contains two disjoint cycles