Disjoint cycles in tournaments and bipartite tournaments

Abstract

A conjecture proposed by Bermond and Thomassen in 1981 states that every digraph with minimum out‐degree at least contains vertex disjoint directed cycles for any integer , which has received substantial attention. This conjecture has been confirmed for . In 2014, Lichiardopol raised a very related conjecture that for every integer , there exists an integer such that every digraph with minimum out‐degree at least contains vertex disjoint directed cycles of different lengths. For a digraph and a set of vertex disjoint directed cycles in , we denote to be the maximum number of directed cycles in of distinct lengths. Let . We define if has no vertex disjoint directed cycles. In this paper, we mainly investigate vertex disjoint directed cycles in tournaments and bipartite tournaments. We first show that for every tournament with minimum out‐degree at least , where . We further prove that for and , any tournament with minimum out‐degree at least satisfies that . Moreover, we deduce that for any tournament with minimum out‐degree at least 7, holds. Additionally, we classify strong bipartite tournaments with minimum out‐degree at least in which any vertex disjoint directed cycles have the same length, where . That is, for any strong bipartite tournament with minimum out‐degree at least , then if and only if is isomorphic to a member of , which is defined in the context.