近生成正则子图
Nearly Spanning Regular Subgraphs
- Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文肯定回答了Alon和Mubayi关于正则图包含近生成正则子图的猜想，对一般k和r给出ε=O_k(r^{-1/2})的结果，并证明k=2时ε=1/(r^2-3)为最优。
AI中文摘要：
Alon和Mubayi曾提出如下问题：对于每个整数$k\ge1$和每个$\varepsilon>0$，是否存在$r_0=r_0(k,\varepsilon)$，使得每个具有$n$个顶点且$r\ge r_0$的$r$-正则图都包含一个覆盖至少$(1-\varepsilon)n$个顶点的$k$-正则子图？此前，该猜想仅在$k\in\{1,2\}$以及$k$和$r$均为偶数的情况下被证明。我们对于一般的$k$和$r$，在$\varepsilon=O_k(r^{-1/2})$的条件下肯定地回答了这个问题。对于$k=2$且每个奇数$r\ge3$，我们证明$\varepsilon=1/(r^2-3)$是充分的，并且这是最优的。
英文摘要：
Alon and Mubayi asked whether, for every integer $k\ge1$ and every $\varepsilon>0$, there exists $r_0=r_0(k,\varepsilon)$ such that every $r$-regular graph on $n$ vertices with $r\ge r_0$ contains a $k$-regular subgraph covering at least $(1-\varepsilon)n$ vertices. Previously, the conjecture was known for $k\in\{1,2\}$ and for $k$ and $r$ both even. We answer this question affirmatively for general $k$ and $r$ with $\varepsilon=O_k(r^{-1/2})$. For $k=2$ and every odd $r\ge3$, we show that $\varepsilon=1/(r^2-3)$ suffices which is best possible.
