Dense circuit graphs and the planar Turán number of a cycle

The planar Turán number ${\text{ex}}_P\left(n,H\right)$ of a graph $H$ is the maximum number of edges in an $n$-vertex planar graph without $H$ as a subgraph. Let $C_k$ denote the cycle of length $k$. The planar Turán number ${\text{ex}}_P\left(n,C_k\right)$ is known for $k\le 7$. We show that dense planar graphs with a certain connectivity property (known as circuit graphs) contain large near triangulations, and we use this result to obtain consequences for planar Turán numbers. In particular, we prove that there is a constant $D$ so that ${\text{ex}}_P(n,C_k)\le 3n-6-Dn/k^{{\log }_2 3}$ for all $k\ge 4$ and $n\ge k^{{\log }_2 3}$. When $k\ge 11$ this bound is tight up to the constant $D$ and proves a conjecture of Cranston, Lidický, Liu, and Shantanam.