Abstract:

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.

Key words: Turán number, planar graph, 5-cycle, 6-cycle