Quasirandom forcing orientations of cycles

This is the authors' abstract. We don't add key points for this paper.

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 is quasirandom-forcing if and only if ≡ 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.