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 and C5 subgraphs that a planar graph on n vertices can contain. We show that K2, n-2 uniquely achieves this maximum in the C4 case, and we identify the graphs which achieve the maximum in the C5 case. This extends work in a paper by...