The planar Turán number of the seven-cycle

The planar Turán number, $e{x}_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $e{x}_{\mathcal{P}}(n,C_4)$ and $e{x}_{\mathcal{P}}(n,C_5)$. Later on, D. Ghosh et al. obtained sharp upper bound of $e{x}_{\mathcal{P}}(n,C_6)$ and proposed a conjecture on $e{x}_{\mathcal{P}}(n,C_k)$ for $k\ge 7$. In this paper, we give a sharp upper bound $e{x}_{\mathcal{P}}(n,C_7)\le \frac{18}{7}n-\frac{48}{7}$, which satisfies the conjecture of D. Ghosh et al. It turns out that this upper bound is also sharp for $e{x}_{\mathcal{P}}(n,\{K_4,C_7\})$, the maximum number of edges in an $n$-vertex planar graph which does not contain $K_4$ or $C_7$ as a subgraph.