A Step toward the Bermond–Thomassen Conjecture about Disjoint Cycles in Digraphs

In 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.