Abstract

AbstractHow many copies of a fixed odd cycle, , can a planar graph contain? We answer this question asymptotically for and prove a bound which is tight up to a factor of 3/2 for all other values of . This extends the prior results of Cox and Martin and of Lv, Győri, He, Salia, Tompkins, and Zhu on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass on the edges of some clique maximizes the probability that edges sampled independently from form either a cycle or a path?

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