Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 0 down vote accepted

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

share|cite|improve this answer
Thanks, I worked through it a year ago, but didn't make the connection. – user12998 Nov 18 '11 at 5:39

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.