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

Abstract

AbstractFor large n we determine exactly the maximum numbers of induced $$C_4$$ C 4 and $$C_5$$ C 5 subgraphs that a planar graph on n vertices can contain. We show that $$K_{2,n-2}$$ K 2 , n - 2 uniquely achieves this maximum in the $$C_4$$ C 4 case, and we identify the graphs which achieve the maximum in the $$C_5$$ 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.