Dense circuit graphs and the planar Turan number of a cycle

The planar Turan number ex(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 Turan number ex(P)(n,C-k) is known for k <= 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 Turan numbers. In particular, we prove that there is a constant D so that ex(P)(n,C-k)<= 3n-6-Dn/k(2)(log) 3 for all k >= 4 and n >= k(2)(log) 3. When k >= 11 this bound is tight up to the constant D and proves a conjecture of Cranston, Lidicky, Liu, and Shantanam.