The maximum number of odd cycles in a planar graph

Abstract

How many copies of a fixed odd cycle, $C_{2m+1}$, can a planar graph contain? We answer this question asymptotically for $m\in \left\{2,3,4\right\}$ 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 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 $\mu$ on the edges of some clique maximizes the probability that $m$ edges sampled independently from $\mu$ form either a cycle or a path?