Planar Turán number of the 7-cycle

Abstract

The planar Turán number ${ex}_P(n,H)$ 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 ${ex}_P(n,C_{\ell })$ is known when $\ell \in \{3,4,5,6\}$, and is expected to behave differently when $\ell \ge 11$. We prove that ${ex}_P(n,C_7)\le \frac{18n}{7}-\frac{48}{7}$ for all $n\ge 39$, and show that equality holds for infinitely many integers $n$.