AI 中文总结
本文开发了稀疏随机图普适性结果的证明框架，确定了二项随机图包含树和环因子的最优条件，回答了Montgomery的问题并改进了相关研究结果。
AI 中文摘要
我们开发了一套用于证明稀疏随机图普适性结果的框架。作为首个应用，我们证明存在绝对常数$C>1$，使得对任意固定常数$\triangle$，二项随机图$G(n,C\fn n/n)$以高概率包含所有最大度不超过$\triangle$的$n$顶点树，这回答了Montgomery（《数学进展》，2019）提出的问题。我们还确定，对所有满足$C\fn n/n \fn n^{-1+o(1)}$的$p$，最小围长$\fn$（相差绝对乘性常数），使得$G(n,p)$以高概率包含所有围长至少为$\fn(\fn)$的环因子。特别地，$G(n,C\fn n/n)$以高概率包含所有围长至少为$100\fn n/\fn\fn n$的环因子，该结果仅相差一个常数因子即为最优。这扩展了Ferber、Kronenberg和Luh（《美国数学会汇刊》，2019）的早期结果，并显著改进了Kahn、Lubetzky和Wormald（《纯粹与应用数学通讯》，2017）的深度结果的一个推论。证明的关键要素之一是确定稀疏随机图中链接系统的最优深度。
英文摘要
We develop a framework for proving universality results in sparse random graphs. As a first application, we show that there exists an absolute constant $C>1$ such that, with high probability, for every fixed constant $Δ$, the binomial random graph $G(n,C\ln n/n)$ contains every $n$-vertex tree with maximum degree at most $Δ$. This answers a question of Montgomery (Advances in Mathematics, 2019). We also determine, for every $p$ satisfying $C\ln n/n\leq p=n^{-1+o(1)}$, the minimum girth $\ell$ (up to an absolute multiplicative constant) for which with high probability $G(n,p)$ contains all cycle factors of girth at least $Ω(\ell)$. In particular, with high probability $G(n,C\ln n/n)$ contains all cycle factors of girth at least $100\ln n/\ln\ln n$, which is optimal up to a constant factor. This extends an earlier result of Ferber, Kronenberg, and Luh (Transaction of the American Mathematical Society, 2019) and significantly improves a corollary of a deep result of Kahn, Lubetzky, and Wormald (Communications on Pure and Applied Mathematics, 2017). One of the key ingredients in the proofs is establishing the optimal depth of linking systems in sparse random graphs.
Comments44 pages, 2 figures, comments are welcome!
