On Tournament Anti-Sidorenko Orientations of Trees

TL;DR

This paper characterizes certain oriented trees called tournament anti-Sidorenko, proving new classes of such trees and paths, confirming conjectures, and expanding understanding of their structure in tournaments.

Contribution

It introduces new classes of tournament anti-Sidorenko oriented trees and paths, proving several conjectures and establishing general conditions for these orientations.

Findings

Paths with specific structural properties are tournament anti-Sidorenko.

All spiders with three legs have a tournament anti-Sidorenko orientation.

The second class of paths is exponentially large with respect to the number of arcs.

Abstract

An oriented graph $\vec{H}$ is said to be tournament anti-Sidorenko if the homomorphism density of $\vec{H}$ in any tournament $\vec{T}$ is bounded above by the homomorphism density of $\vec{H}$ in a large uniformly random tournament. We prove the following: (1) Every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko. (2) An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a multiple of four. (3) Every spider with exactly three legs admits a tournament anti-Sidorenko orientation. The first result proves a conjecture posed by He, Mani, Nie, Tung and Wei. The third resolves a problem from the same paper, in fact establishing a substantially more general statement, and provides evidence in support of a conjecture of Fox, Himwich, Mani and Zhou. The second yields the first family of tournament anti-Sidorenko oriented paths which is exponentially large with respect to the number of arcs.