Let $F$ be a graph. The planar Turan number of $F$, denoted by $text{ex}_{mathcal{P}}(n,F)$, is the maximum number of edges in an $n$-vertex planar graph containing no copy of $F$ as a subgraph. Let $Theta_k$ denote the family of Theta graphs on $kgeq 4$ vertices, that is, a graph obtained by joining a pair of non-consecutive vertices of a $k$-cycle with an edge. Y. Lan, et.al. determined sharp upper bound for $text{ex}_{mathcal{P}}(n,Theta_4)$ and $text{ex}_{mathcal{P}}(n,Theta_5)$. Moreover, they obtained an upper bound for $text{ex}_{mathcal{P}}(n,Theta_6)$. They proved that, $text{ex}_{mathcal{P}}(n,Theta_6)leq frac{18}{7}n-frac{36}{7}$. In this paper, we improve their result by giving a bound which is sharp. In particular, we prove that $text{ex}_{mathcal{P}}(n,Theta_6)leq frac{18}{7}n-frac{48}{7}$ and demonstrate that there are infinitely many $n$ for which there exists a $Theta_6$-free planar graph $G$ on $n$ vertices, which attains the bound.
تحميل البحث