Planar Graphs with the Maximum Number of Induced 4-Cycles or 5-Cycles

For large n we determine exactly the maximum numbers of induced C₄ C 4 and C₅ C 5 subgraphs that a planar graph on n vertices can contain. We show that K2,n-2 K 2 , n - 2 uniquely achieves this maximum in the C₄ C 4 case, and we identify the graphs which achieve the maximum in the C₅ 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.