论文标题
长度四的循环严格为$ f $-turán-good
The cycle of length four is strictly $F$-Turán-good
论文作者
论文摘要
给定不包含$ f $作为子图的$（r+1）$ - 色度图$ f $和图$ h $，我们说$ h $严格是$ f $ -f $-turán-good如果turán图$ t_ {r}（r}（n）$是包含$ h $ f $ n $ $ f $ f $ f $ f $ f $ f $ f $ f $ f $ n $ n $ f $ f $ f $ f $ n $ n $ f $ f $ f $ n $ n $ n $ n $ n $ fertice的唯一图。 Győri，Pach and Simonovits（1991）证明了四个循环$ c_4 $长度为$ k_ {r+1} $ - turán-good for All $ r \ geq 2 $。在本文中，我们扩展了此结果，并表明$ C_4 $严格是$ f $-turán-good，其中$ f $是$（r+1）$ - $ r \ ge 2 $和颜色临界边缘的色度图。此外，我们表明，每一个$ n $ vertex $ c_4 $ -free Graph $ g $带有$ n（h，g）= \ ex（n，c_4，f）-o（n^4）$可以通过添加或删除$ o（n^2）$ edges $ o（n^2）$。我们的证明使用Razborov（2007）开发的FLAG代数方法。
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. Győri, Pach and Simonovits (1991) proved that cycle $C_4$ of length four is strictly $K_{r+1}$-Turán-good for all $r\geq 2$. 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 $C_4$-free graph $G$ with $N(H,G)=\ex(n,C_4,F)-o(n^4)$ 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 (2007).
