Vertex-disjoint cycles of different lengths in tournaments

Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least 2k-1 contains k vertex-disjoint cycles, here k is a positive integer. Lichiardopol conjectured in 2014 that for every positive integer k there exists an integer g (k) such that every digraph with minimum outdegree at least g (k) contains k vertex-disjoint cycles of different lengths. Recently, Chen and Chang proved in J. Graph Theory 105 (2) (2024) 297-314 that for k 3 every tournament with minimum outdegree at least 2k-1 contains k vertex-disjoint cycles in which two of them have different lengths. Motivated by the above two conjectures and related results, we investigate vertex-disjoint cycles of different lengths in tournaments, and show that when k 5 every tournament with minimum outdegree at least 2k-1 contains k vertex-disjoint cycles in which three of them have different lengths. In addition, we show that every tournament with minimum outdegree at least 6 contains three vertex-disjoint cycles of different lengths and the minimum outdegree condition is sharp. This answers a question proposed by Chen and Chang.