اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدPlanar Turan Number of the $Theta_6$
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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.
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Let ${rm ex}_{mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${rm ex}_{mathcal{P}}(n,T,H)$ is the well studied function, the planar Turan number of
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The Turan number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices which does not contain $H$ as a subgraph. Let $P_{k}$ denote the path on $k$ vertices and let $mP_{k}$ denote $m$ disjoint copies of $P
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