Variations on Sidorenko's conjecture in tournaments
arxiv(2024)
摘要
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))klog_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.
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