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.