Planar graphs with the maximum number of induced 4-cycles or 5-cycles

For large \(n\) we determine exactly the maximum numbers of induced \(C_4\) and \(C_5\) subgraphs that a planar graph on \(n\) vertices can contain. We show that \(K_{2,n-2}\) uniquely achieves this maximum in the \(C_4\) case, and we identify the graphs which achieve the maximum in the \(C_5\) case. This extends work in a paper by Hakimi and Schmeichel and a paper by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora which together determine both maxima asymptotically.