Two proofs of Bermond-Thomassen conjecture for regular tournaments

Bermond-Thomassen conjecture says that a digraph of minimum out-degree at least 2r−1, r⩾1, contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r=2, but it is still open for larger values of r, even when restricted to (regular) tournaments. In this paper, we present two proofs of this conjecture for regular tournaments. In the first one, we shall prove auxiliary results about union of sets contained in other union of sets, that might be of independent interest. The second one uses a more graph-theoretical approach, by studying the properties of a maximum set of vertex-disjoint directed triangles.