Planar Turán Numbers of Cycles: A Counterexample

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.