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A sharp asymptotic bound for odd cycles in planar graphs · arXivDesk

2608.13162Aug 13, 20269 pages

A sharp asymptotic bound for odd cycles in planar graphs

Zhen Liu, Chuanshu Wu

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 $\mathbf{N}_{\mathcal P}(n,H)$

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For graphs G and H, let N(G,H) denote the number of unlabeled, not necessarily induced copies of H in G, and let N_P(n,H) be the maximum of N(G,H) over all n-vertex planar graphs G. We prove that, for every fixed integer m≥ 3, N_P(n,C₂m+1) =2m((n/m))^m +O_m (n^m-1/5).

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Claims

The paper addresses: A sharp asymptotic bound for odd cycles in planar graphs.
For graphs G and H, let N(G,H) denote the number of unlabeled, not necessarily induced copies of H in G, and let N_P(n,H) be the maximum of N(G,H) over all n-vertex planar graphs G.

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Nearby in the stack

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).

Heath, Martin, and Wells reduced the determination of the leading term to a weighted optimization conjecture involving cycles and paths. We prove a stronger sharp cycle--path inequality for probability weights on the edges of a complete graph and characterize equality in their conjectured inequality. Together with their reduction lemma, this settles the conjecture and yields the formula above, including the stated error term. The cases

m\geq 5

are new; combined with the known results for

C_3

and

C_5

, this determines the leading term for every fixed odd cycle in planar graphs.

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