渗滤稠密图中的弹性森林普适性
Resilient forest universality in percolated dense graphs
- The Taft School（塔夫特学校）
- Wesleyan University（卫斯理安大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文证明了一个密度敏感的转移定理，将稀疏随机图中的弹性转化为稠密图的森林普适性，并推广了 Erdős-Sós 猜想的近似形式在随机边删除下的稳健性。
中文摘要 AI 辅助
Christoph、Müyesser 和 Wigderson 最近提出一个问题：Erdős-Sós 猜想的一个近似形式是否在随机边删除下具有稳健性。我们建立了一个密度敏感的转移定理，将稀疏随机图中有界度树的全局弹性转化为任意稠密宿主图中的弹性森林普适性。更精确地说，若 $F$ 是一个 $N$ 顶点图，其边密度 $λ$ 远离零，且 $p\in[K/N,1]$，则对于宿主图和渗滤参数一致地以 $1-o(1)$ 的概率，从 $F_p$ 中删除至多 $α$ 比例的边后得到的每个子图都包含所有至多 $((1-α)λ-ξ)N$ 个顶点的有界度森林。因此，对于每个 $c>0$、$L\ge1$、固定的 $D$ 和 $α<c$，每个至多有 $Ld$ 个顶点且平均度至少为 $d$ 的图 $F$ 具有如下性质：在任意速率 $p\in[K/d,1]$ 下渗滤后，再删除至多 $α$ 比例的幸存边，剩余图对所有至多 $(1-c)d$ 个顶点且最大度至多 $D$ 的森林是普适的。这包括任意指定集合的有界度树的总阶数的顶点不相交打包。我们进一步获得了连通分量的顶点覆盖数至多为 $Cd$ 的图，以及通过在 $dn$ 尺度上删除足够少的边可转化为这种形式的图的森林普适性结果。
英文摘要
Christoph, Müyesser and Wigderson recently asked whether an approximate form of the Erdős-Sós conjecture is robust under random edge deletions. We establish a density-sensitive transference theorem that converts global resilience for bounded-degree trees in sparse random graphs into resilient forest universality in arbitrary dense host graphs. More precisely, if $F$ is an $N$-vertex graph of edge density $λ$ bounded away from zero and $p\in[K/N,1]$, then, with probability $1-o(1)$ uniformly over the host and the percolation parameter, every subgraph obtained from $F_p$ by deleting at most an $α$-fraction of its edges contains every bounded-degree forest on at most $((1-α)λ-ξ)N$ vertices. Consequently, for every $c>0$, $L\ge1$, fixed $D$ and $α<c$, every graph $F$ on at most $Ld$ vertices with average degree at least $d$ has the property that, after percolation at any rate $p\in[K/d,1]$ and any subsequent deletion of at most an $α$-fraction of the surviving edges, the remaining graph is universal for all forests on at most $(1-c)d$ vertices and maximum degree at most $D$. This includes vertex-disjoint packings of any prescribed collection of bounded-degree trees of that total order. We further obtain forest-universality results for graphs whose connected components have vertex-cover number at most $Cd$, and for graphs that can be brought into this form by deleting sufficiently few edges on the $dn$-scale.
