On the Number of Disjoint 4-Cycles in Regular Tournaments - Ma Fuhong (Discussiones Mathematicae Graph Theory, 2018)
In this paper, we prove that for an integer r ≥ 1, every regular tournament T of degree 3r − 1 contains at least 2116r-103${{21} \over {16}}r - {{10} \over 3}$ disjoint directed 4-cycles. Our result is an improvement of Lichiardopol’s theorem when taking q = 4 [Discrete Math. 310 (2010) 2567–2570]: for given integers q ≥ 3 and r ≥ 1, a tournament T with minimum out-degree and in-degree both at least (q − 1)r − 1 contains at least r disjoint directed cycles of length q.