arXiv 1408.54882014-08math.COcs.DM

超立方体的饱和与Bootstrap渗流

Saturation in the Hypercube and Bootstrap Percolation

Natasha Morrison, Jonathan A. Noel, Alex Scott

备注：: 21 pages, 2 figures. To appear in Combinatorics, Probability and Computing
DOI：: 10.1017/S0963548316000122

摘要：

令$Q_d$表示维数为$d$的超立方体。给定$d\geq m$，$Q_d$的生成子图$G$称为$(Q_d,Q_m)$ -饱和，如果它不包含$Q_m$作为子图，但添加$E(Q_d)\setminus E(G)$的任何边都会在$G$中创建$Q_m$的副本。本文讨论了Johnson和Pinto提出的一个问题，证明了对于每个固定的$m\geq2$，$(Q_d,Q_m)$ -饱和图的最小边数是$Θ(2^d)$ .我们还研究了弱饱和，这是一种形式的自助渗流。如果$E(Q_d)\setminus E(G)$的边可以一次一个地添加到$G$中，使得每个添加的边创建$Q_m$的新副本，则称$Q_d$的生成子图是弱$(Q_d,Q_m)$ -饱和的。我们讨论了Johnson和Pinto的另一个问题，确定了弱$(Q_d,Q_m)$ -饱和图的最小边数，对所有的$d\geq m\geq1$ .更一般地说，我们确定的最小数量的边缘在一个子图的$d$维网格$P_k^d$是弱饱和的“轴对齐”副本的一个较小的网格$P_r^m$。我们还研究了网格中的弱饱和圈。

英文摘要：

Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $Θ(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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