# $3$-cycles in regular tournaments

A regular tournament is a complete digraph on $n$ vertices such that in-degree and out-degree of each vertex is equal to $\dfrac{n-1}{2}$. A locally-transitive regular tournament is a regular tournament with the additional property that the in-neighborhood and the out-neighborhood of each vertex forms a transitive tournament of order $\dfrac{n-1}{2}$.

It is straightforward to show that each vertex in a locally-transitive regular tournament lies on exactly $\dfrac{n^2-1}{8}$ distinct $3$-cycles. More generally, this property also seems to hold for all regular tournaments, however, I do not have a proof of this statement.

I would appreciate any insight into whether or not we should expect this property to hold for all regular tournaments. Also, I would be happy to read through any references that you could point me towards that are at least tangentially related to this topic.

Take a look at page $9$ of John Moon's book, Topics in Tournaments, Holt, Rinehart and Winston, 1968.