Extremal \(C_{4}\)-free/\(C_{5}\)-free planar graphs (Q2833118)

The authors study the topic of ``extremal'' planar graphs, defining \(\mathrm{ex}_p(n,H)\) 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, the case when \(H\) is a small cycle is investigated, obtaining \(\operatorname{ex}_p(n,C_4)\leq \frac{15}{7}(n-2)\) for all \(n\geq 4\) and \(\operatorname{ex}_p(n,C_5)\leq \frac{12n-33}{5}\) for all \(n\geq 11\), and showing that both of these bounds are tight.