Planar Tur\'an Number of the ${\mathrm{\Theta }}_6$

Let $\mathcal{F}$ be a nonempty family of graphs. A graph $G$ is called $\mathcal{F}$-$\mathit{\text{free}}$ if it contains no graph from $\mathcal{F}$ as a subgraph. For a positive integer $n$, the \emph{planar Tur'an number} of $\text{F}$, denoted by ${\text{ex}}_{\text{p}}(n,\text{F})$, is the maximum number of edges in an $n$-vertex $\text{F}$-free planar graph. Let ${\mathrm{\Theta }}_k$ be the family of Theta graphs on $k\ge 4$ vertices, that is, graphs obtained by joining a pair of non-consecutive vertices of a $k$-cycle with an edge. Lan, Shi and Song determined an upper bound ${\text{ex}}_{\mathcal{P}}(n,{\mathrm{\Theta }}_6)\le \frac{18}{7}n-\frac{36}{7}$, but for large $n$, they did not verify that the bound is sharp. In this paper, we improve their bound by proving $$\text{ex}$_{$\mathcal{P}$}(n,\Theta_6)\leq \frac{18}{7}n-\frac{48}{7}$ and then we demonstrate the existence of infinitely many positive integer $n$ and an $n$-vertex ${\mathrm{\Theta }}_6$-free planar graph attaining the bound.