发表机构
South China Normal University（华南师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明仅由 Cheeger 常数下界即可保证图存在大小为 Ω(√(nd/log d)) 的完全 minor，并改进 d-正则图的收缩团数下界。
AI 中文摘要
设 $G$ 是一个有 $n$ 个顶点的图，并设 $e_G(S,V(G)\setminus S)$ 是恰好有一个端点在 $S$ 中的边的数量。$G$ 的 Cheeger 常数和限制 Cheeger 常数（其中 $k\ge1$ 是实数）分别为 \\[ h(G)=\min_{\substack{\emptyset\ne S\subseteq V(G)\\\\|S|\le \frac{n}{2}}} \frac{e_G(S,V(G)\setminus S)}{|S|} \text{ 和 } h_k(G)=\min_{\substack{\emptyset\ne S\subseteq V(G) |S|\le\min\{k,\frac{n}{2}\}}} \frac{e_G(S,V(G)\setminus S)}{|S|}. \\] 收缩团数 $\ccl(G)$ 是最大的整数 $r$，使得 $G$ 包含完全图 $K_r$ 作为一个 minor。Krivelevich 和 Nenadov [Complete minors in graphs without sparse cuts, Int. Math. Res. Not. IMRN 12 (2021) 8996--9015] 证明了，对于每个固定的 $\eps>0$ 和所有足够大的 $n$ 和 $d$，如果 $G$ 是一个有 $n$ 个顶点且最大度至多为 $d$ 的图，那么 $h(G)\ge\eps d$ 和 $h_{\eps n}(G)\ge(\frac{1}{2}+\eps)d$ 蕴含 $\ccl(G)=\Omega_\eps(\sqrt{\frac{nd}{\log d}})$。他们询问是否可以确定在不需要关于 $h_{\eps n}(G)$ 的额外条件的情况下，能否保证 $\ccl(G)$ 的相同下界。他们证明了当 $d$ 是常数时情况确实如此。我们肯定地回答了这个问题。对于每个 $\eps>0$，存在常数 $\beta=\beta(\eps)>0$ 和 $n_0=n_0(\eps)$，使得只要 $d\ge 3$ 是整数，对于每个有 $n\ge n_0$ 个顶点且最大度至多为 $d$ 的图 $G$，如果 $h(G)\ge\eps d$，那么 $\ccl(G)\ge\beta\sqrt{\frac{nd}{\log d}}$。这个下界对 $n$ 和 $d$ 的依赖在常数因子意义下是最优的。作为推论，对于第二大的特征值远离 $d$ 的 $d$-正则图，我们导出了其收缩团数的下界，与之前的 $\frac{d}{2}$ 相比有所改进。证明结合了图的谱性质与惰性随机游走的分析。
英文摘要
Let $G$ be a graph on $n$ vertices, and let $e_G(S,V(G)\setminus S)$ be the number of edges with exactly one endpoint in $S$. The Cheeger constant and the restricted Cheeger constant of $G$, where $k\ge1$ is real, are, respectively, \[ h(G)=\min_{\substack{\emptyset\ne S\subseteq V(G)\\|S|\le \frac{n}{2}}} \frac{e_G(S,V(G)\setminus S)}{|S|} \text{ and } h_k(G)=\min_{\substack{\emptyset\ne S\subseteq V(G) |S|\le\min\{k,\frac{n}{2}\}}} \frac{e_G(S,V(G)\setminus S)}{|S|}. \] The contraction clique number $\ccl(G)$ is the largest integer $r$ such that $G$ contains the complete graph $K_r$ as a minor. Krivelevich and Nenadov [Complete minors in graphs without sparse cuts, Int. Math. Res. Not. IMRN 12 (2021) 8996--9015] proved that, for every fixed $\eps>0$ and all sufficiently large $n$ and $d$, if $G$ is a graph on $n$ vertices with maximum degree at most $d$, then $h(G)\ge\eps d$ and $h_{\eps n}(G)\ge(\frac{1}{2}+\eps)d$ imply $\ccl(G)=Ω_\eps(\sqrt{\frac{nd}{\log d}})$. They asked to determine if one can guarantee the same lower bound on $\ccl(G)$ without the additional condition on $h_{\eps n}(G)$. They showed that this is the case when $d$ is a constant. We answer this question affirmatively. For every $\eps>0$, there are constants $β=β(\eps)>0$ and $n_0=n_0(\eps)$ such that, whenever $d\ge 3$ is an integer, for every graph $G$ with $n\ge n_0$ vertices and maximum degree at most $d$, if $h(G)\ge\eps d$, then $\ccl(G)\geβ\sqrt{\frac{nd}{\log d}}$. The dependence of this lower bound on $n$ and $d$ is best possible up to a constant factor. As a corollary, a lower bound is derived for the contraction clique number of $d$-regular graphs for which the second largest eigenvalue is bounded away from $d$, compared to earlier $\frac{d}{2}$. The proof combines spectral properties of graphs with an analysis of lazy random walks.
