Vertex-disjoint cycles of different lengths in tournaments

Abstract

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 $k\geqslant 3$ 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 $k\geqslant 5$ 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.