On the number of 4-cycles in a tournament

TL;DR

This paper investigates the relationship between the number of 3-cycles and 4-cycles in tournaments, proposing a conjecture about the structure minimizing 4-cycles and deriving bounds using flag algebras.

Contribution

It introduces the first study of inducibility in tournaments and provides bounds on 4-cycle counts based on 3-cycle density, using flag algebra methods.

Findings

Derived a lower bound on the number of 4-cycles close to the conjectured minimum.

Solved the problem of maximizing the number of 4-cycles given 3-cycle density.

Proposed that extremal tournaments resemble random blow-ups of transitive tournaments.

Abstract

If $T$ is an $n$-vertex tournament with a given number of $3$-cycles, what can be said about the number of its $4$-cycles? The most interesting range of this problem is where $T$ is assumed to have $c\cdot n^3$ cyclic triples for some $c>0$ and we seek to minimize the number of $4$-cycles. We conjecture that the (asymptotic) minimizing $T$ is a random blow-up of a constant-sized transitive tournament. Using the method of flag algebras, we derive a lower bound that almost matches the conjectured value. We are able to answer the easier problem of maximizing the number of $4$-cycles. These questions can be equivalently stated in terms of transitive subtournaments. Namely, given the number of transitive triples in $T$, how many transitive quadruples can it have? As far as we know, this is the first study of inducibility in tournaments.