On the number of 5-cycles in a tournament

TL;DR

This paper derives an exact formula for counting 5-cycles in tournaments, establishes bounds, and shows that most tournaments contain nearly the maximum possible number of such cycles, matching the expected count in random tournaments.

Contribution

It provides a precise formula for 5-cycle counts in tournaments and demonstrates that the maximum number aligns with the expected count in random tournaments, highlighting typical tournament structure.

Findings

Maximum 5-cycle count asymptotically equals  in a random tournament.

Almost all tournaments contain the maximum number of 5-cycles.

The formula relates 5-cycle counts to the edge score sequence.

Abstract

We find an exact formula for the number of directed 5-cycles in a tournament in terms of its edge score sequence. We use this formula to find both upper and lower bounds on the number of 5-cycles in any $n$-tournament. In particular, we show that the maximum number of 5-cycles is asymptotically equal to $\frac{3}{4}{n \choose 5}$, the expected number 5-cycles in a random tournament ($p=\frac{1}{2}$), with equality (up to order of magnitude) for almost all tournaments. Note that this means that almost all $n$-tournaments contain the maximum number of $5$-cycles.