树退化图和嵌套依赖随机选择

T. Jiang, Sean Longbrake
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引用次数: 0

摘要

著名的依赖随机选择引理指出,在二部图中,一个平均顶点(按其度加权)具有这样的性质,即在其邻域中几乎所有的小子集$S$都具有与具有相同边密度的随机图中几乎一样大的公共邻域。引理的两个著名应用如下。第一个是f redi和Alon、Krivelevich和Sudakov的定理,该定理表明,在不包含最大度为$r$的固定二部图的$n$顶点图中,边的最大个数为$O(n^{2-1/r})$。最近,Grzesik, Janzer和Nagy把这个扩展到所谓的$(r,t)$ -一棵树的膨胀。第二个应用是Conlon, Fox和Sudakov的一个定理,证实了Erd \H{o} s和Simonovits以及Sidorenko猜想的一个特例,表明如果$H$是一个包含一个顶点完备于另一部分的二部图,并且$G$是一个图,那么从$V(H)$到$V(G)$的一致随机映射至少$\left[\frac{2|E(G)|}{|V(G)|^2}\right]^{|E(H)|}$是同态的概率。在这篇文章中,我们引入了依赖随机选择引理的一个嵌套变体,它可能会引起独立的兴趣。然后,我们应用它来获得Conlon, Fox, and Sudakov定理和Grzesik, Janzer, and Nagy定理关于Turán和所谓的树退化图的Sidorenko性质的共同扩展。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Tree-Degenerate Graphs and Nested Dependent Random Choice
The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets $S$ in its neighborhood has common neighborhood almost as large as in the random graph of the same edge-density. Two well-known applications of the lemma are as follows. The first is a theorem of F\"uredi and of Alon, Krivelevich, and Sudakov showing that the maximum number of edges in an $n$-vertex graph not containing a fixed bipartite graph with maximum degree at most $r$ on one side is $O(n^{2-1/r})$. This was recently extended by Grzesik, Janzer and Nagy to the family of so-called $(r,t)$-blowups of a tree. A second application is a theorem of Conlon, Fox, and Sudakov, confirming a special case of a conjecture of Erd\H{o}s and Simonovits and of Sidorenko, showing that if $H$ is a bipartite graph that contains a vertex complete to the other part and $G$ is a graph then the probability that the uniform random mapping from $V(H)$ to $V(G)$ is a homomorphismis at least $\left[\frac{2|E(G)|}{|V(G)|^2}\right]^{|E(H)|}$. In this note, we introduce a nested variant of the dependent random choice lemma, which might be of independent interest. We then apply it to obtain a common extension of the theorem of Conlon, Fox, and Sudakov and the theorem of Grzesik, Janzer, and Nagy, regarding Tur\'an and Sidorenko properties of so-called tree-degenerate graphs.
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