Erdős-Gallai定理的鲁棒性与超稳定性
Robustness and hyperstability for the Erdős-Gallai theorem
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中文总结 AI 辅助
证明Erdős-Gallai定理的鲁棒扩展：平均度d的图以概率p独立渗流后，渐近几乎必然包含长度至少(1-c)d的环；并证明超稳定版本：无长环的图可通过至多c d n次边删除转化为所有连通分支顶点覆盖大小至多(1+c)d的图。
中文摘要 AI 辅助
Erdős-Gallai定理指出，每个平均度为$d$的图都包含一个长度至少为$d$的环。我们证明了Erdős-Gallai定理的以下鲁棒扩展：对于每个$c>0$，存在$K$，使得对于所有$d\geq K$，$p\geq K/d$以及每个平均度为$d$的图$G$，通过以概率$p$独立渗流$G$的每条边得到的随机图$G_p$，当$|V(G)|\to\infty$时，渐近几乎必然包含一个长度为$(1-c)d$的环。利用相关方法，我们证明了Erdős-Gallai定理的以下超稳定版本：任何没有长度至少为$d$的环的图$G$，最多通过$c d n$次边删除，就可以转化为一个所有连通分支都具有大小为$(1+c)d$的顶点覆盖的图。我们论证的核心是一个关于图的非常一般的结构定理，该定理源于Pokrovskiy关于有界度树超稳定性的结果。
英文摘要
The Erdős--Gallai theorem states that every graph of average degree $d$ contains a cycle of length at least $d$. We prove the following robust extension of the Erdős--Gallai theorem: For every $c>0$ there exists $K$ such that for all $d\geq K$, $p\geq K/d$ and every graph $G$ with average degree $d$, the random graph $G_p$ obtained by independently sampling each edge of $G$ with probability $p$ contains a cycle of length at least $(1-c)d$ asymptotically almost surely as $|V(G)|\to \infty$. With related methods, we prove the following hyperstability version of the Erdős--Gallai theorem: any graph $G$ without a cycle of length at least $d$ is at most $c d|V(G)|$ edge deletions away from a graph all of whose connected components have a vertex-cover of size at most $d$. At the core of our argument lies a very general structure theorem about graphs that originates from results of Pokrovskiy concerning the hyperstability of bounded-degree trees.
