Extremal C4-free/C5-free planar graphs

We study the topic of "extremal" planar graphs, defining $\mathrm{ex_{_{\mathcal{P}}}}(n,H)$ to be the maximum number of edges possible in a planar graph on $n$ vertices that does not contain a given graph $H$ as a subgraph. In particular,we examine the case when $H$ is a small cycle,obtaining $\mathrm{ex_{_{\mathcal{P}}}}(n,C_{4}) \leq \frac{15}{7}(n-2)$ for all $n \geq 4$ and $\mathrm{ex_{_{\mathcal{P}}}}(n,C_{5}) \leq \frac{12n-33}{5}$ for all $n \geq 11$, and showing that both of these bounds are tight.