Geodesic
On the Number of Disjoint 4-Cycles in Regular Tournaments
Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 491-498

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In this paper, we prove that for an integer r ≥ 1, every regular tournament T of degree 3r − 1 contains at least 2116 r- 103 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.
Keywords: regular tournament, C 4 -free, disjoint cycles 
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Ma, Fuhong; Yan, Jin. On the Number of Disjoint 4-Cycles in Regular Tournaments. Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 2, pp. 491-498. http://geodesic.mathdoc.fr/item/DMGT_2018_38_2_a11/