Dense circuit graphs and the planar Turán number of a cycle

Abstract

The $\text{planar Turaˊn number}$ ${\text{ex}}_{\mathcal{P}}(n,H)$ 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 ${\text{ex}}_{\mathcal{P}}(n,C_k)$ 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 ${\text{ex}}_{\mathcal{P}}(n,C_k)\le 3n-6-Dn/k^{{\log }_2^3}$ for all $k,n\ge 4$. When $k\ge 11$ this bound is tight up to the constant $D$ and proves a conjecture of Cranston, Lidický, Liu, and Shantanam.