Planar Tur\'{a}n Numbers of Cycles: A Counterexample

TL;DR

This paper disproves a conjecture about the maximum edges in large planar graphs avoiding cycles of length at least 11, providing counterexamples and proposing revised conjectures.

Contribution

It provides the first counterexamples to a longstanding conjecture on planar Turán numbers for cycles of length at least 11.

Findings

Disproved the conjecture for all 0-cycle cases

Constructed explicit counterexamples for 0-cycle graphs

Proposed two revised conjectures to better describe the maximum edges in such graphs.

Abstract

The planar Turan number $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ is the largest number of edges in an $n$-vertex planar graph with no $\ell$-cycle. For $\ell\in \{3,4,5,6\}$, upper bounds on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ are known that hold with equality infinitely often. Ghosh, Gy\"{o}ri, Martin, Paulo, and Xiao [arxiv:2004.14094] conjectured an upper bound on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ for every $\ell\ge 7$ and $n$ sufficiently large. We disprove this conjecture for every $\ell\ge 11$. We also propose two revised versions of the conjecture.