Variations on Sidorenko's conjecture in tournaments

TL;DR

This paper investigates variants of Sidorenko's conjecture in tournaments, revealing that certain oriented graphs are either sparse or over-represented, and characterizes specific orientations like stars.

Contribution

It introduces the concepts of tournament anti-Sidorenko and Sidorenko graphs, providing bounds, constructions, and characterizations within tournament orientations.

Findings

Tournament anti-Sidorenko graphs are very sparse, with edges at most about k log k.

Most edge deletions from transitive tournaments produce Sidorenko graphs.

Characterization of star orientations as Sidorenko or anti-Sidorenko.

Abstract

We study variants of Sidorenko's conjecture in tournaments, where new phenomena arise that do not have clear analogues in the setting of undirected graphs. We first consider oriented graphs that are systematically under-represented in tournaments (called tournament anti-Sidorenko). We prove that such oriented graphs must be quite sparse; specifically, the maximum number of edges of a $k$-vertex oriented graph which is tournament anti-Sidorenko is $(1+o(1))k\log_2 k$. We also give several novel constructions of oriented graphs that are systematically over-represented in tournaments (tournament Sidorenko); as a representative example, we show that most ways to delete an edge from a transitive tournament yield a tournament Sidorenko oriented graph. As an illustration of our methods, we characterize which orientations of stars are tournament Sidorenko and which are tournament anti-Sidorenko.