The Cycle of Length Four is Strictly F-Turán-Good

Abstract

Given an $(r+1)$-chromatic graph F and a graph H that does not contain F as a subgraph, we say that H is strictly F-Turán-good if the Turán graph $T_{r}(n)$ is the unique graph containing the maximum number of copies of H among all F-free graphs on n vertices for every n large enough. Györi et al. (Graphs Comb 7(1):31–37, 1991) proved that cycle $C_4$ of length four is strictly $K_{r+1}$-Turán-good for all $r\ge 2$. In this article, we extend this result and show that $C_4$ is strictly F-Turán-good, where F is an $(r+1)$-chromatic graph with $r\ge 2$ and a color-critical edge. Moreover, we show that every n-vertex F-free graph G with $N(C_4,G)=\text{ ex }(n,C_4,F)-o(n^4)$ can be obtained by adding or deleting $o(n^2)$ edges from $T_r(n)$. Our proof uses the flag algebra method developed by Razborov (J Symb Logic 1239–1282, 2007).