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arXiv · 2212.09343

Quasirandom forcing orientations of cycles

Andrzej Grzesik·Daniel Il'kovič·Bartłomiej Kielak·Daniel Král'
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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.

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Andrzej Grzesik, Daniel Il'kovič, Bartłomiej Kielak, Daniel Král'. 2022-12-19. Quasirandom forcing orientations of cycles. https://arxiv.org/abs/2212.09343

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