设$\mathcal{H}$是一个图族。$\mathcal{H}$的平面Turán数, 记为$\text{ex}_{\mathcal{P}}(n,\mathcal{H})$, 表示不包含$\mathcal{H}$中任一个图的n阶平面图的最大边数。本文研究图的特定子图-三角块的划分, 在该划分基础上对每个三角块对图G的顶点、边和面的贡献计数。结合平面图的结构特性并通过双向计数和归纳技巧得到如下结果: 设G是禁用$\{C_5,C_6\}$的连通n阶平面图, 如果$n \geq 14$, 则$e(G)\leq\frac{30n-84}{13}$。在此基础上, 构造了无穷多个达到该界值的极图。

Let $\mathcal{H}$ be a family of graphs. The planar Turán number of $\mathcal{H}$, denoted by ex$_{\mathcal{P}}(n,\mathcal{H})$, is the maximum number of edges over all planar graphs on n vertices that do not contain any $H\in\mathcal{H}$ as a subgraph. In this paper, we partition the graph into some specific subgraphs called as triangle blocks. Based on the partition, we calculate the contribution of each triangle block to the vertex, edge and face to graph G, respectively. Considered the structure of plane graph and by double-counting techniques and induction on the order of G, we obtained the following result: Let G be a connected $\{C_5,C_6\}$-free plane graph on n vertices, if $n\geq14$, then $e(G)\leq\frac{30n-84}{13}$. We also give infinitely many graphs which attain these bounds.