New Sidorenko-type inequalities in tournaments

TL;DR

This paper investigates Sidorenko-type inequalities in tournaments, focusing on oriented trees and cycles, and provides new results, characterizations, and algorithms related to the Sidorenko property in directed graphs.

Contribution

It establishes new inequalities for oriented trees and cycles, advances the understanding of Sidorenko properties in tournaments, and offers algorithms for characterizing these properties in paths and cycles.

Findings

Progress on the conjecture that all trees have an anti-Sidorenko direction

Characterization of short paths regarding Sidorenko properties

Algorithms for determining Sidorenko status of paths and cycles

Abstract

As a directed analog of Sidorenko's conjecture in extremal graph theory, Fox, Himwich, Zhou, and the second author defined an oriented graph $H$ to be tournament Sidorenko (anti-Sidorenko) if the random tournament asymptotically minimizes (maximizes) the number of copies of $H$ among all tournaments. We prove new inequalities of this form for oriented trees and cycles, considering both local and global notions of the Sidorenko property. We make progress on a conjecture of the aforementioned authors that every tree has an anti-Sidorenko direction, and give a characterization of short paths. For long paths we show that orientations are split symmetrically between being locally Sidorenko and anti-Sidorenko, yet almost all orientations are not globally Sidorenko. Finally, we give algorithms characterizing the local Sidorenko status of paths and cycles when the number of vertices is not divisible by four.