On 5-cycles and strong 5-subtournaments in a tournament of odd order n

Let T be a tournament of odd order n≥ 5, cₘ(T) be the number of its m-cycles, and sₘ(T) be the number of its strongly connected m-subtournaments. Due to work of L.W. Beineke and F. Harary, it is well known that sₘ(T)≤ sₘ(RLTₙ), where RLTₙ is the regular locally transitive tournament of order $n.$ For $m=3$ and $m=4,$ cₘ(T) equals sₘ(T), but it is not so for m≥ 5. As J.W. Moon pointed out in his note in 1966, the problem of determining the maximum of cₘ(T) seems very difficult in general (i.e. for m≥ 5). In the present paper, based on the Komarov-Mackey formula for c₅(T) obtained recently, we prove that c₅(T)≤ (n+1)n(n-1)(n-2)(n-3)/160 with equality holding iff T is doubly regular. A formula for s₅(T) is also deduced. With the use of it, we show that s₅(T)≤ (n+1)n(n-1)(n-3)(11n-47)/1920 with equality holding iff T=RLTₙ or $n=7$ and T is regular or $n=5$ and T is strong. It is also proved that for a regular tournament T of (odd) order n≥ 9, a lower bound (n+1)n(n-1)(n-3)(17n-59)/3840≤ s₅(T) holds with equality iff T is doubly regular. These results are compared with the ones recently obtained by the author for c₅(T).