New Sidorenko-type inequalities in tournaments

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.

💡 Research Summary

This paper investigates Sidorenko-type inequalities within the constrained setting of tournaments, offering a directed analog of the famous Sidorenko conjecture in extremal graph theory. A directed graph H is defined as tournament Sidorenko (TS) if the random tournament asymptotically minimizes the number of copies of H among all tournaments of the same size, and tournament anti-Sidorenko (TAS) if it asymptotically maximizes this count.

The core contributions are multifaceted, bridging spectral theory, combinatorial analysis, and algorithmic characterization.

Methodological Foundation: The authors employ a novel polynomial expansion and spectral method. By decomposing a tournament’s adjacency matrix into a quasirandom component (1/2 J) and a structured skew-symmetric perturbation (B), they derive expressions for homomorphism densities. This allows for the analysis of orientations previously intractable, leading to a complete classification of all oriented paths up to length 5 and providing evidence for a conjecture that paths with a single direction flip are TAS.

Local vs. Global Phenomena: A key insight is the distinction between local and global Sidorenko properties. A graph is locally TS/TAS if the inequality holds for all sufficiently quasirandom tournaments. For oriented paths and cycles, the paper provides combinatorial characterizations of these local properties based on the signs of signed counts of certain subgraphs (e.g., C(P3, D)). An efficient algorithm is given to classify the local status of paths when the number of vertices is not divisible by four. It is shown that for long paths, orientations are split symmetrically between being locally TS and locally TAS.

In stark contrast, the study of global properties reveals a different landscape. Theorem 1.8 proves that the fraction of oriented paths that are globally TS is vanishingly small (o_n(1)), implying that the random tournament is almost never the minimizer for homomorphism density of a long path. This is complemented by a conjecture that a large proportion (1/2 - o_n(1)) of orientations are instead globally TAS.

Progress on the TAS Tree Conjecture: The paper makes significant progress on a conjecture that every undirected tree admits a TAS orientation. By introducing the concept of an isomorphic pair, Theorem 1.3 provides a reduction method, showing that if a tree contains such a pair and the remaining subgraphs have TAS orientations, then the whole tree does. Furthermore, Theorem 1.4 proves that all caterpillar trees have a TAS orientation. These results substantially narrow the class of trees for which the conjecture remains open.

Overall Significance: This work establishes a comprehensive framework for studying extremal problems in tournaments. It demonstrates that the directed, tournament-constrained version of Sidorenko’s problem exhibits rich and often counterintuitive behavior not present in the undirected case, particularly in the dichotomy between local and global properties. The techniques developed—spectral expansions, local analysis via subgraph counts, and probabilistic arguments for long paths—provide a robust toolkit for future research in this area.