Cycles of a given length in tournaments

TL;DR

This paper investigates the maximum number of directed cycles of a fixed length in tournaments, establishing exact values and asymptotic behavior, and confirming a conjecture about when this maximum matches the expected count in random tournaments.

Contribution

It proves that the maximum cycle count ratio equals one if and only if the cycle length isn't divisible by four, and provides bounds and exact values for specific lengths.

Findings

c(3)=1, c(4)=4/3

c(ell)=1 if and only if ell not divisible by 4

Exact value c(8)=2 for cycles of length 8

Abstract

We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let $c(\ell)$ be the limit of the ratio of the maximum number of cycles of length $\ell$ in an $n$-vertex tournament and the expected number of cycles of length $\ell$ in the random $n$-vertex tournament, when $n$ tends to infinity. It is well-known that $c(3)=1$ and $c(4)=4/3$. We show that $c(\ell)=1$ if and only if $\ell$ is not divisible by four, which settles a conjecture of Bartley and Day. If $\ell$ is divisible by four, we show that $1+2\cdot\left(2/\pi\right)^{\ell}\le c(\ell)\le 1+\left(2/\pi+o(1)\right)^{\ell}$ and determine the value $c(\ell)$ exactly for $\ell = 8$. We also give a full description of the asymptotic structure of tournaments with the maximum number of cycles of length $\ell$ when $\ell$ is not divisible by four or $\ell\in\{4,8\}$.