论文标题
$θ_6$的平面图
Planar Turán Number of the $Θ_6$
论文作者
论文摘要
令$ \ mathcal {f} $为非空的图。图形$ g $称为$ \ Mathcal {f} $ - \ textit {free}，如果它包含$ \ Mathcal {f} $的图形为子图。对于一个正整数$ n $，$ \ f $的\ emph {planarturán数字}，由$ \ ex _ {\ p}表示（n，\ f）$表示，是$ n $ vertex $ \ f f $ fume planar图中的最大边缘。 令$θ_k$为$ k \ geq 4 $顶点的theta图的家族，也就是说，通过连接一对具有边缘的$ k $ cycle的非连续顶点获得的图形。 LAN，Shi和Song确定了上限$ \ text {ex} _ {\ Mathcal {p}}（n，θ_6）\ leq \ leq \ frac {18} {7} n- \ frac {36} {36} {7} {7} {7} $，但对于大的$ n $，他们没有验证范围。在本文中，我们通过证明$ \ text {ex} _ {\ mathcal {p}}（n，θ_6）\ leq \ leq \ frac {18} {7} n- \ frac {48} {7} {7} $，然后我们证明了无限的inte $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $平面图达到了界限。
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 Turán number} of $\F$, denoted by $\ex_{\p}(n,\F)$, is the maximum number of edges in an $n$-vertex $\F$-free planar graph. Let $Θ_k$ 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 $\text{ex}_{\mathcal{P}}(n,Θ_6)\leq \frac{18}{7}n-\frac{36}{7}$, but for large $n$, they did not verify that the bound is sharp. In this paper, we improve their bound by proving $\text{ex}_{\mathcal{P}}(n,Θ_6)\leq \frac{18}{7}n-\frac{48}{7}$ and then we demonstrate the existence of infinitely many positive integer $n$ and an $n$-vertex $Θ_6$-free planar graph attaining the bound.
