The maximum number of odd cycles in a planar graph

TL;DR

This paper determines the maximum number of odd cycles in planar graphs for small cycle lengths and provides bounds for larger lengths, extending previous work on even cycles and connecting to a probabilistic maximum likelihood problem.

Contribution

It extends the understanding of odd cycle counts in planar graphs and introduces a novel reduction to a maximum likelihood problem involving edge probability distributions.

Findings

Asymptotic bounds for the number of odd cycles in planar graphs for m=2,3,4

Tight bounds up to a factor of 3/2 for other cycle lengths

Connection between cycle counts and a maximum likelihood edge distribution problem

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?