Weak diameter and cyclic properties in oriented graphs

We describe several conditions on the minimum number of arcs ensuring that any two vertices in a strong oriented graph are joining by a path of length at most a given $k$, or ensuring that they are contained in a common cycle

Edge disjoint Hamiltonian cycles in highly connected tournaments

Thomassen conjectured that there is a function f(k) such that every strongly f(k)-connected tournament contains k edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kühn, Lapinskas, Osthus, and Patel who showed that f(k) ≤ O(k 2 (logk) 2 ) and conjectured that there is a constant C such that f(k) ≤ Ck 2 . We prove this conjecture. As a second application of our methods we answer a question of Thomassen about spanning linkages in highly connected tournaments