论文标题
在尺寸和三个方面的三个邻居引导渗透的极端界限
Extremal Bounds for Three-Neighbour Bootstrap Percolation in Dimensions Two and Three
论文作者
论文摘要
对于$ r \ geq1 $，图$ g $中的$ r $ -neighbour bootstrap进程从一组受感染的顶点开始，在每个时间步骤中，每个顶点至少具有$ r $ $ $ $ $ $ $ $ $的感染邻居感染。如果$ g $的每个顶点最终都被感染，则最初的感染会渗透。当$ g $是$ 3 $二维的电网时，我们确切地确定了$ 3 $ -Neighbour Bootstrap流程渗透的套装的最低基数，最低侧面长度至少$ 11 $。我们还表征了整个Integers $ a $ a $ a和$ b $，其中有一组基数$ \ frac {ab+a+a+a+a+b} {3} $，在$ 3 $ -neighbour bootstrap过程中渗透了$ a \ times b $ b $ grid;这解决了Benevides，Bermond，Lesfari和Nisse提出的问题[HAL研究报告03161419V4，2021]。
For $r\geq1$, the $r$-neighbour bootstrap process in a graph $G$ starts with a set of infected vertices and, in each time step, every vertex with at least $r$ infected neighbours becomes infected. The initial infection percolates if every vertex of $G$ is eventually infected. We exactly determine the minimum cardinality of a set that percolates for the $3$-neighbour bootstrap process when $G$ is a $3$-dimensional grid with minimum side-length at least $11$. We also characterize the integers $a$ and $b$ for which there is a set of cardinality $\frac{ab+a+b}{3}$ that percolates for the $3$-neighbour bootstrap process in the $a\times b$ grid; this solves a problem raised by Benevides, Bermond, Lesfari and Nisse [HAL Research Report 03161419v4, 2021].
