Minimizing cycles in tournaments and normalized $q$-norms

TL;DR

This paper investigates the minimization of cycles of various lengths in tournaments, extending previous work on 3- and 4-cycles, and introduces new results on the asymptotic behavior of cycle counts for different cycle lengths.

Contribution

It proves asymptotic minimization results for cycles of certain lengths in tournaments and addresses a question on $q$-norm minimization for probabilistic vectors, expanding understanding of cycle structures.

Findings

For cycle lengths not congruent to 2 mod 4, extremal tournaments minimize cycle counts when density is high.

Confirmed the minimization of 4-cycle counts for $d extgreater 1/36$ using spectrum analysis.

Constructed explicit tournaments with fewer cycles of lengths congruent to 2 mod 4, countering the minimization trend.

Abstract

Akin to the Erd\H{o}s-Rademacher problem, Linial and Morgenstern made the following conjecture in tournaments: for any $d\in (0,1]$, among all $n$-vertex tournaments with $d\binom{n}{3}$ many 3-cycles, the number of 4-cycles is asymptotically minimized by a special random blow-up of a transitive tournament. Recently, Chan, Grzesik, Kr\'al' and Noel introduced spectrum analysis of adjacency matrices of tournaments in this study, and confirmed this for $d\geq 1/36$. In this paper, we investigate the analogous problem of minimizing the number of cycles of a given length. We prove that for integers $\ell\not\equiv 2\mod 4$, there exists some constant $c_\ell>0$ such that if $d\geq 1-c_\ell$, then the number of $\ell$-cycles is also asymptotically minimized by the same family of extremal examples for $4$-cycles. In doing so, we answer a question of Linial and Morgenstern about minimizing the $q$-norm of a probabilistic vector with given $p$-norm for any integers $q>p>1$. For integers $\ell\equiv 2\mod 4$, however the same phenomena do not hold for $\ell$-cycles, for which we can construct an explicit family of tournaments containing fewer $\ell$-cycles for any given number of $3$-cycles. We conclude by proposing two conjectures on the minimization problem for general cycles in tournaments.