Planar Turán number of the 7-cycle

Abstract

The $\text{planar Turaˊn number}$ ${\text{ex}}_{\mathcal{P}}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_{\ell }$ denote the cycle of length $\ell$. The planar Turán number ${\text{ex}}_{\mathcal{P}}(n,C_{\ell })$ behaves differently for $\ell \le 10$ and for $\ell \ge 11$, and it is known when $\ell \in \{3,4,5,6\}$. We prove that ${\text{ex}}_{\mathcal{P}}(n,C_7)\le \frac{18n}{7}-\frac{48}{7}$ for all $n>38$, and show that equality holds for infinitely many integers $n$.