Sidorenko猜想关于细分与theta替换
Sidorenko's conjecture for subdivisions and theta substitutions
AI总结：
本文证明了用由偶路径构成的广义theta图替换任意图的边（在满足特定整除条件时）所得图满足Sidorenko猜想，并无条件证明了完全图的此类替换二部图满足该猜想，推广了Conlon等人的结果。
摘要：
著名的Sidorenko猜想断言，对于每个二部图$H$，在给定边密度的条件下，当$G$为伪随机图时，从$H$到图$G$的同态数量达到最小。我们证明，对于任意图$H$，若用由偶路径构成的广义theta图替换$H$的边，在路径数量满足特定整除条件时，所得图满足Sidorenko猜想。为此，我们无条件地证明了：将完全图每条边替换为广义theta图所得的二部图满足Sidorenko猜想，这推广了Conlon、Kim、Lee和Lee [J. Lond. Math. Soc., 2018] 的结果。
英文摘要：
The famous Sidorenko's conjecture asserts that for every bipartite graph $H$, the number of homomorphisms from $H$ to a graph $G$ with given edge density is minimized when $G$ is pseudorandom. We prove that for any graph $H$, a graph obtained from replacing edges of $H$ by generalized theta graphs consisting of even paths satisfies Sidorenko's conjecture, provided a certain divisibility condition on the number of paths. To achieve this, we prove unconditionally that bipartite graphs obtained from replacing each edge of a complete graph with a generalized theta graph satisfy Sidorenko's conjecture, which extends a result of Conlon, Kim, Lee and Lee [J. Lond. Math. Soc., 2018].
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