Cycles of a given length in tournaments

Abstract

We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let c(ℓ) be the limit of the ratio of the maximum number of cycles of length ℓ in an n-vertex tournament and the expected number of cycles of length ℓ 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(ℓ)=1 if and only if ℓ is not divisible by four, which settles a conjecture of Bartley and Day. If ℓ is divisible by four, we show that 1+2⋅(2/π)ℓ≤c(ℓ)≤1+(2/π+o(1))ℓ and determine the value c(ℓ) exactly for ℓ=8. We also give a full description of the asymptotic structure of tournaments with the maximum number of cycles of length ℓ when ℓ is not divisible by four or ℓ∈{4,8}.