An improvement of Lichiardopol’s theorem on disjoint cycles in tournaments

摘要

Let k ≥ 1 and q ≥ 3 be integers and let f(q)=(6q2−16q+10)/(3q2−3q−4). In this paper, we prove that if q ≥ 4, then every tournament T with both minimum out-degree and in-degree at least (q−1)k−1 contains at least f(q)k−2q disjoint cycles of length q. We also prove that if q=3 and k ≥ 6, then T contains at least 16k/15−5 disjoint triangles. Our results improve Lichiardopol’s theorem ([Discrete Math. 310 (19) (2010) 2567–2570]): for given integers q ≥ 3 and k ≥ 1, a tournament T with both minimum out-degree and in-degree at least (q−1)k−1 contains at least k disjoint cycles of length q.