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

Abstract

Let $T$ be a tournament of odd order $n\ge 5,$ $c_m(T)$ be the number of its $m$-cycles, and $s_m(T)$ be the number of its strongly connected $m$-subtournaments. Due to work of L.W. Beineke and F. Harary, it is well known that $s_m(T)\le s_m(RL{T}_n),$ where $RL{T}_n$ is the regular locally transitive tournament of order $n.$ For $m=3$ and $m=4,$ $c_m(T)$ equals $s_m(T),$ but it is not so for $m\ge 5.$ As J.W. Moon pointed out in his note in 1966, the problem of determining the maximum of $c_m(T)$ seems very difficult in general (i.e. for $m\ge 5$). In the present paper, based on the Komarov-Mackey formula for $c_5(T)$ obtained recently, we prove that $c_5(T)\le (n+1)n(n-1)(n-2)(n-3)/160$ with equality holding iff $T$ is doubly regular. A formula for $s_5(T)$ is also deduced. With the use of it, we show that $s_5(T)\le (n+1)n(n-1)(n-3)(11n-47)/1920$ with equality holding iff $T=RL{T}_n$ 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 $n\ge 9,$ a lower bound $(n+1)n(n-1)(n-3)(17n-59)/3840\le s_5(T)$ holds with equality iff $T$ is doubly regular. These results are compared with the ones recently obtained by the author for $c_5(T).$