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arXiv 0710.2296math.COcs.DMmath.PR
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扩张图的点渗透

Vertex Percolation on Expander Graphs

Sonny Ben-Shimon, Michael Krivelevich

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13 pages
发表信息：
European Journal of Combinatorics, 30(2), pp. 339-350, 2009
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摘要：

我们称$n$顶点上的图$G=(V,E)$是$β>0$的$β$ -扩张子，如果每个基数为$|U|\leq \frac{n}{2}$的$U\subseteq V$满足$|N_G(U)|\geq β|U|$，其中$N_G(U)$表示$U$的邻域.本文研究了当$β$ -扩张器的顶点数为$α>0$时，以概率$n^{-α}$独立随机删除的过程，并研究了所得到的图的性质.我们的主要结果指出，$n$趋于无穷大，删除过程中执行的$β$ -扩展图的有界度将导致高概率的图形组成的一个巨大的组件包含$n-o(n)$顶点，这本身就是一个扩展图，和恒定大小的组件。我们继续通过应用的主要结果扩展图具有正的谱隙。在$(n,d,λ)$ -图（即此类扩展器）的特定情况下，我们在图上的附加约束下计算$α$的值，对于这些约束，所得图很有可能保持连接，或者由巨型分量和孤立顶点组成。由于从$d$ -正则图的均匀概率空间中采样的具有高概率的图是扩展图并且满足附加约束，因此这一结果加强了Greenhill，Holt和Wormald关于随机$d$ -正则图的顶点渗流的最近结果.我们的结论是，通过示出在以无界扩展比扩展子线性集的图上执行上述删除过程，以高概率产生连通扩展图。

英文摘要：

We say that a graph $G=(V,E)$ on $n$ vertices is a $β$-expander for some constant $β>0$ if every $U\subseteq V$ of cardinality $|U|\leq \frac{n}{2}$ satisfies $|N_G(U)|\geq β|U|$ where $N_G(U)$ denotes the neighborhood of $U$. In this work we explore the process of deleting vertices of a $β$-expander independently at random with probability $n^{-α}$ for some constant $α>0$, and study the properties of the resulting graph. Our main result states that as $n$ tends to infinity, the deletion process performed on a $β$-expander graph of bounded degree will result with high probability in a graph composed of a giant component containing $n-o(n)$ vertices that is in itself an expander graph, and constant size components. We proceed by applying the main result to expander graphs with a positive spectral gap. In the particular case of $(n,d,λ)$-graphs, that are such expanders, we compute the values of $α$, under additional constraints on the graph, for which with high probability the resulting graph will stay connected, or will be composed of a giant component and isolated vertices. As a graph sampled from the uniform probability space of $d$-regular graphs with high probability is an expander and meets the additional constraints, this result strengthens a recent result due to Greenhill, Holt and Wormald about vertex percolation on random $d$-regular graphs. We conclude by showing that performing the above described deletion process on graphs that expand sub-linear sets by an unbounded expansion ratio, with high probability results in a connected expander graph.

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