Extremal Results on Disjoint Cycles in Tournaments and Bipartite Tournaments

In this paper, we give two extremal results on vertex disjoint-directed cycles in tournaments and bipartite tournaments. Let $q\ge 2$ and $k\ge 2$ be two integers. The first result is that for every strong tournament $D$, with a minimum out-degree of at least $\left(q-1\right)k-1$ with $q\ge 3$, any $k$ vertex disjoint-directed cycle, which has a length of at least $q$ in $D$, has the same length if and only if $q=3,k=2$ and $D$ is isomorphic to $P{T}_7$. The second result is that for each strong bipartite tournament $D$, with a minimum out-degree of at least $qk-1$ with $q$ being even, any $k$ vertex disjoint-directed cycle, each of which has a length of at least $2q$ in $D$, has the same length if and only if $D$ is isomorphic to a member of $BT\left(n_1,n_2,\dots ,n_{qk}\right)$. Our results generalize some results of Tan and of Chen and Chang, and in a sense, extend several results of Bang-Jensen et al. of Ma et al. as well as of Wang et al.