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arXiv · 2307.00116

The maximum number of odd cycles in a planar graph

Emily Heath·Ryan R. Martin·Chris Wells
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Abstract

How many copies of a fixed odd cycle, $C_{2m+1}$, can a planar graph contain? We answer this question asymptotically for $m\in\{2,3,4\}$ and prove a bound which is tight up to a factor of $3/2$ for all other values of $m$. This extends the prior results of Cox--Martin and Lv et al. on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass $\mu$ on the edges of some clique maximizes the probability that $m$ edges sampled independently from $\mu$ form either a cycle or a path?

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Emily Heath, Ryan R. Martin, Chris Wells. 2023-06-30. The maximum number of odd cycles in a planar graph. https://arxiv.org/abs/2307.00116

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