Disjoint cycles in tournaments and bipartite tournaments

A conjecture proposed by Bermond and Thomassen in 1981 states that every digraph with minimum out-degree at least 2k-1 contains k vertex disjoint directed cycles for any integer k >= 1, which has received substantial attention. This conjecture has been confirmed for k <= 3. In 2014, Lichiardopol raised a very related conjecture that for every integer k >= 2, there exists an integer g(k) such that every digraph with minimum out-degree at least g(k) contains k vertex disjoint directed cycles of different lengths. For a digraph D and a set of k vertex disjoint directed cycles C in D, we denote kappa(k)(C) to be the maximum number of directed cycles in C of distinct lengths. Let kappa(k)(D)=max{kappa(k)(C)vertical bar C is a set of k vertex disjoint directed cycles inD}. We define kappa(k)(D) = 0 if D has no k vertex disjoint directed cycles. In this paper, we mainly investigate vertex disjoint directed cycles in tournaments and bipartite tournaments. We first show that kappa(k)(D) >= 2 for every tournament D with minimum out-degree at least 2k - 1, where k >= 3. We further prove that for k >= 1 and gamma is an element of{1,2, ..., k}, any tournament D with minimum out-degree at least gamma(2)-2 gamma+6k-3/2 satisfies that kappa(k)(D) >= gamma. Moreover, we deduce that for any tournament D with minimum out-degree at least 7, kappa(3)(D)=3 holds. Additionally, we classify strong bipartite tournaments with minimum out-degree at least 2k-1 in which any k vertex disjoint directed cycles have the same length, where k >= 2. That is, for any strong bipartite tournament D with minimum out-degree at least 2k-1, then kappa(k)(D)=1 if and only if D is isomorphic to a member of BT(n(1),n(2), ... ,n(2)k), which is defined in the context.