Cycles in a tournament with pairwise zero, one or two given vertices in common

Then, for a connected tournament of diameter at most 3, we determine a relation between the maximum number of vertex-disjoint cycles and the maximum number of vertex-disjoint cycles spanning the vertex set of T. Also, by using a lemma of Chen et al. [Partitioning vertices of a tournament into independent cycles, J. Combin. Theory Ser. B 83 (2001) 213-220], we prove that a k-connected tournament of order at least 5k-3, of diameter distinct from 3 (resp. 3) admits k (resp. k-1) vertex-disjoint cycles spanning the vertex set of T, with only one exception. Finally, we give results on cycles with pairwise one or two vertices in common. A few open problems are raised.