Let $\mathcal{ F}$ be a nonempty family of graphs. A graph $G$ is called $\mathcal{ F}$-\textit{ free} if it contains no graph from $\mathcal{ F}$ as a subgraph. For a positive integer $n$, the \emph{ planar Tur\'an number} of $\F$, denoted by $\ex_{ \p}(n,\F)$, is the maximum number of edges in an $n$-vertex $\F$-free planar graph. Let $\Theta_k$ be the family of Theta graphs on $k\geq 4$ 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 $\text{ ex}_{ \mathcal{ P}}(n,\Theta_6)\leq \frac{ 18}{ 7}n-\frac{ 36}{ 7}$, 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 $\Theta_6$-free planar graph attaining the bound.
Categories: math.CO