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 $\varphi\colon V(H)\to V(G)$ so that $\varphi(u)\varphi(v)\in E(G)$ 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 $(p,\lambda)$-bijumbled graphs of minimum degree which is a constant proportion of the average degree, i.e.\ $\Omega(pn)$. A $(p,\lambda)$-bijumbled graph is characterised through the discrepancy property: $\left|e(A,B)-p|A||B|\right |<\lambda\sqrt{|A||B|}$ for any two sets of vertices $A$ and $B$. Our condition $\lambda=O(p^2n/\log n)$ 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 is based on that of Nenadov (\emph{Bulletin of the London Mathematical Society}, to appear) and on ours (arXiv:1806.01676), together with additional ideas and simplifications.

中文翻译：

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在稀疏伪随机图中有效地找到任何给定的 2-factor

给定一个$n$-顶点伪随机图$G$和一个最大度数最多为2的$n$-顶点图$H$，我们希望在$G$中找到$H$的副本，即\一个嵌入$ \varphi\colon V(H)\to V(G)$ 使得 $\varphi(u)\varphi(v)\in E(G)$ 对于所有 $uv\in E(H)$。此问题的特定实例包括在 $G$ 中找到三角形因子和找到哈密顿圈。在这里，我们提供了一个确定性多项式时间算法，它可以在任何合适的伪随机图 $G$ 中找到给定的 $H$。我们考虑的伪随机图是最小度数的 $(p,\lambda)$-bijumbled 图，它是平均度数的恒定比例，即 $\Omega(pn)$。$(p,\lambda)$-bijumbled 图的特征在于差异性： $\left|e(A,B)-p|A||B|\right |<\lambda\sqrt{|A|| B|}$ 表示任意两组顶点 $A$ 和 $B$。我们关于混杂性的条件 $\lambda=O(p^2n/\log n)$ 处于紧的对数因子内，并为最近 Nenadov 的问题提供了肯定的答案。我们结合了吸收储层方法的新变体，这是一种来自极值图论和随机图的强大工具。我们的方法基于 Nenadov（\emph{伦敦数学会公报}，出现）和我们的方法 (arXiv:1806.01676)，以及其他想法和简化。