爆破的Sidorenko猜想
Sidorenko's conjecture for blow-ups
摘要：
Sidorenko和Erdens-Simonovits的一个著名猜想指出，对于所有的二部图$H$，拟随机图渐近地包含在所有具有相同阶和边密度的图上所取的$H$的最小拷贝数。这个猜想在过去的十年里引起了相当大的兴趣，现在已知它适用于广泛的二部图，总体趋势是，如果一个图可以用简单的构建块（如树）以某种递归的方式构建，那么它就满足这个猜想。我们的贡献在这里，这超出了这个范例，是表明，猜想持有任何二部图$H$与二划分$A \cup B$，在$B$的度$k$的顶点数满足一定的整除条件，为每个$k$。作为推论，我们得到：对于任何具有二分图$A \cup B$的二部图$H$，存在正整数$p$，使得取$H$的$p$顶点不相交副本，沿相应顶点粘合$A$的所有副本所形成的爆破$H_A^p$满足猜想.另一种观察后一结果的方法是，对于每个二分$H$，都有一个正整数$p$，使得Sidorenko猜想的$L^p$版本对$H$成立。
英文摘要：
A celebrated conjecture of Sidorenko and Erdős-Simonovits states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion. Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph $H$ with bipartition $A \cup B$ where the number of vertices in $B$ of degree $k$ satisfies a certain divisibility condition for each $k$. As a corollary, we have that for every bipartite graph $H$ with bipartition $A \cup B$, there is a positive integer $p$ such that the blow-up $H_A^p$ formed by taking $p$ vertex-disjoint copies of $H$ and gluing all copies of $A$ along corresponding vertices satisfies the conjecture. Another way of viewing this latter result is that for every bipartite $H$ there is a positive integer $p$ such that an $L^p$-version of Sidorenko's conjecture holds for $H$.
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