正则图中几乎独立集的计数

中文摘要 AI 辅助

Kahn证明了，在n个顶点的二部d-正则图中，独立集的数量由K_{d,d}副本的不相交并集取得最大值。Zhao后来将此结果推广到所有d-正则图。我们证明了该定理的一个稳健版本，其中独立集被替换为跨越少量内部边的集合。若G是n个顶点的d-正则图，则跨越至多γdn条边的子集数量至多为2^{n/2}exp{O(γlog(e/γ)n)+O(n/d)}。两个修正项在绝对常数意义下都是精确的：当d相对于γ足够大时，γlog(1/γ)n项是必要的，而n/d项对于独立集已经是必要的。我们的结果回答了Seth的一个问题。

英文摘要

Kahn proved that, among bipartite $d$-regular graphs on $n$ vertices, the number of independent sets is maximized by a disjoint union of copies of $K_{d,d}$. Zhao later extended this result to all $d$-regular graphs. We prove a robust version of this theorem in which independent sets are replaced by sets spanning few internal edges. If $G$ is $d$-regular on $n$ vertices, then the number of subsets spanning at most $γdn$ edges is at most $$ 2^{n/2}\exp\left\{ O\bigl(γ\log(e/γ)n\bigr) + O\bigl(n/d\bigr) \right\}. $$ Both correction terms are sharp up to absolute constants: the $γ\log(1/γ)n$ term is necessary when $d$ is sufficiently large in terms of $γ$, while the $n/d$ term is already necessary for independent sets. Our result answers a question of Seth.

发表机构

• Trinity College Cambridge（剑桥大学三一学院）

机构由 AI 辅助整理，请以论文原文为准。