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arXiv · 2605.14138

On Tournament Anti-Sidorenko Orientations of Trees

Hao Chen·Felix Christian Clemen·Jonathan A. Noel
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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.

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Hao Chen, Felix Christian Clemen, Jonathan A. Noel. 2026-05-13. On Tournament Anti-Sidorenko Orientations of Trees. https://arxiv.org/abs/2605.14138

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