Quasirandom forcing orientations of cycles

TL;DR

This paper investigates which orientations of cycles in tournaments are quasirandom-forcing, establishing conditions for even cycles and classifying orientations of cycles up to length 10.

Contribution

It generalizes known results on cyclic orientations, provides necessary and sufficient conditions for quasirandom-forcing of cycle orientations, and classifies small cycle orientations.

Findings

Cyclic orientation of length $ ext{mod } 4$ is quasirandom-forcing.

No odd cycle orientation is quasirandom-forcing.

Classified orientations of cycles up to length 10 as quasirandom-forcing or not.

Abstract

An oriented graph $H$ is quasirandom-forcing if the limit (homomorphism) density of $H$ in a sequence of tournaments is $2^{-\|H\|}$ if and only if the sequence is quasirandom. We study generalizations of the following result: the cyclic orientation of a cycle of length $\ell$ is quasirandom-forcing if and only if $\ell\equiv 2$ mod $4$. We show that no orientation of an odd cycle is quasirandom-forcing. In the case of even cycles, we find sufficient conditions on an orientation to be quasirandom-forcing, which we complement by identifying necessary conditions. Using our general results and spectral techniques used to obtain them, we classify which orientations of cycles of length up to $10$ are quasirandom-forcing.