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).
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Acknowledgements
We thank Professor Dániel Gerbner for remarks on the history of the problem of F-Tuán-good and valuable discussions on this manuscript.
Funding
The work was supported by the National Natural Science Foundation of China (No. 12071453) and the National Key R and D Program of China(2020YFA0713100), the Anhui Initiative in Quantum Information Technologies (AHY150200) and the Innovation Program for Quantum Science and Technology, China (2021ZD0302902)
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Hei, D., Hou, X. The Cycle of Length Four is Strictly F-Turán-Good. Bull. Malays. Math. Sci. Soc. 47, 5 (2024). https://doi.org/10.1007/s40840-023-01602-2
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DOI: https://doi.org/10.1007/s40840-023-01602-2