给定一个$n$顶点伪随机图$G$和一个最大度至多为2的$n$顶点图$H$，我们希望在$G$中找到$H$的一个副本，即一个嵌入$\varphi\colon V(H)\to V(G)$，使得对于所有$uv\in E(H)$有$\varphi(u)\varphi(v)\in E(G)$。该问题的具体实例包括在$G$中找到三角形因子和哈密顿环。这里，我们提供了一个确定性多项式时间算法，可以在任何适当伪随机的图$G$中找到给定的$H$。我们考虑的伪随机图是最小度为平均度常数比例（即$\Omega(pn)$）的$(p,\lambda)$-双扰动图。$(p,\lambda)$-双扰动图通过差异性质刻画：对于任意两个顶点集$A$和$B$，有$\left|e(A,B)-p|A||B|\right|<\lambda\sqrt{|A||B|}$。我们对双扰动度的条件$\lambda=O(p^2n/\log n)$在log因子内是紧的，并对Nenadov最近的问题给出了肯定回答。我们结合了吸收-储层方法的新变体，这是极值图论和随机图中的一个强大工具。我们的方法基于我们之前的工作（\emph{European Journal of Combinatorics} \textbf{82} (2019), 102999），并融入了Nenadov的工作（\emph{Bulletin of the London Mathematical Society} \textbf{51} (3) (2019), pp.~421--430），以及额外的想法和简化。
英文摘要：
Given an $n$-vertex pseudorandom graph $G$ and an $n$-vertex graph $H$ with maximum degree at most two, we wish to find a copy of $H$ in $G$, i.e.\ an embedding $φ\colon V(H)\to V(G)$ so that $φ(u)φ(v)\in E(G)$ for all $uv\in E(H)$. Particular instances of this problem include finding a triangle-factor and finding a Hamilton cycle in $G$. Here, we provide a deterministic polynomial time algorithm that finds a given $H$ in any suitably pseudorandom graph $G$. The pseudorandom graphs we consider are $(p,λ)$-bijumbled graphs of minimum degree which is a constant proportion of the average degree, i.e.\ $Ω(pn)$. A $(p,λ)$-bijumbled graph is characterised through the discrepancy property: $\left|e(A,B)-p|A||B|\right |<λ\sqrt{|A||B|}$ for any two sets of vertices $A$ and $B$. Our condition $λ=O(p^2n/\log n)$ on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov.
We combine novel variants of the absorption-reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work (\emph{European Journal of Combinatorics} \textbf{82} (2019), 102999), incorporating the work of Nenadov (\emph{Bulletin of the London Mathematical Society} \textbf{51} (3) (2019), pp.~421--430), together with additional ideas and simplifications.
Counting results for sparse pseudorandom hypergraphs I
我们建立了一个所谓的计数引理，它允许将某些线性均匀超图嵌入到稀疏伪随机超图中，推广了图的结果[Embedding graphs with bounded degrees in sparse pseudorandom graphs，Israel J.Math.139（2004），93-137]。我们的结果的应用在配套文件[稀疏伪随机超图计数结果II]。
Hamilton cycles in pseudorandom graphs: resilience and approximate decompositions
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Department of Mathematical Sciences, KAIST（韩国科学技术院数学科学系）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极值组合与概率研究组）
