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

Chen et al. [Partitioning vertices of a tournament into independent cycles, J. Combin. Theory Ser. B 83 (2001) 213–220] proved that every k-connected tournament with at least 8k vertices admits k vertex-disjoint cycles spanning the vertex set, which answered a question posed by Bollobas. In this paper, we prove, as a consequence of a more general result, that every k-connected tournament of diameter at least 4 contains k vertex-disjoint cycles spanning the vertex set. 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. © 2007 Elsevier B.V. All rights reserved.