On Tournament Anti-Sidorenko Orientations of Trees
Hao Chen, Felix Christian Clemen, Jonathan A. Noel

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 is said to be tournament anti-Sidorenko if the homomorphism density of in any tournament is bounded above by the homomorphism density of 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,…
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