A sharp asymptotic bound for odd cycles in planar graphs

Zhen Liu; Chuanshu Wu

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arXiv · 2608.13162

A sharp asymptotic bound for odd cycles in planar graphs

Zhen Liu·Chuanshu Wu

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Abstract

For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1})=2m\left(\frac{n}{m}\right)^m+O_m\!\left(n^{m-1/5}\right).$$ The proof uses a sharp weighted cycle--path inequality for edge probability measures on finite complete graphs. This strengthens a conjecture of Heath, Martin, and Wells and, together with their reduction lemma, yields the stated asymptotic formula.

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Zhen Liu, Chuanshu Wu. 2026-08-13. A sharp asymptotic bound for odd cycles in planar graphs. https://arxiv.org/abs/2608.13162

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