The maximum number of 10- and 12-cycles in a planar graph

TL;DR

This paper determines the maximum number of 10- and 12-cycles in large planar graphs, extending known results for smaller cycles, and explores probabilistic models to identify optimal edge distributions for cycle formation.

Contribution

It establishes asymptotic counts for the maximum number of 10- and 12-cycles in planar graphs and investigates probability measures that maximize cycle formation likelihood.

Findings

Asymptotic maximum of (n/5)^5 for 10-cycles

Asymptotic maximum of (n/6)^6 for 12-cycles

Optimal edge probability measures for cycle formation

Abstract

For a fixed planar graph $H$, let $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In the case when $H$ is a cycle, the asymptotic value of $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_m)$ is currently known for $m\in\{3,4,5,6,8\}$. In this note, we extend this list by establishing $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{10})\sim(n/5)^5$ and $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{12})\sim(n/6)^6$. We prove this by answering the following question for $m\in\{5,6\}$, which is interesting in its own right: which probability mass $\mu$ on the edges of some clique maximizes the probability that $m$ independent samples from $\mu$ form an $m$-cycle?