We introduce a simple method for proving lower bounds for the size of the smallest percolating set in a certain graph bootstrap process. We apply this method to obtain recursive formulas for the sizes of the smallest percolating sets in multidimensional tori and multidimensional grids (in particular hypercubes). The former answers a question of Morrison and Noel [9], and the latter provides an alternative and simpler proof for one of their main results.

中文翻译：

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通过多项式的性质进行图自举渗透的下界

我们介绍了一种简单的方法，用于证明某个图引导过程中最小渗滤集的大小的下界。我们应用此方法获得多维托里和多维网格（特别是超立方体）中最小渗滤集的大小的递归公式。前者回答了Morrison和Noel [9]的问题，而后者为其主要结果之一提供了另一种更简单的证明。