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Percolating sets in bootstrap percolation on the Hamming graphs and triangular graphs
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2020-10-21 , DOI: 10.1016/j.ejc.2020.103256
Mohammadreza Bidgoli , Ali Mohammadian , Behruz Tayfeh-Rezaie

The r-neighbor bootstrap percolation on a graph is an activation process of the vertices. The process starts with some initially activated vertices and then, in each round, any inactive vertex with at least r active neighbors becomes activated. A set of initially activated vertices leading to the activation of all vertices is said to be a percolating set. Denote the minimum size of a percolating set in the r-neighbor bootstrap percolation process on a graph G by m(G,r). In this paper, we present upper and lower bounds on m(Knd,r), where Knd is the Cartesian product of d copies of the complete graph Kn which is referred as the Hamming graph. Among other results, when d goes to infinity, we show that m(Knd,r)=1+o(1)(d+1)!rd if rd2 and nr+1. Furthermore, we explicitly determine m(L(Kn),r), where L(Kn) is the line graph of Kn also known as triangular graph.



中文翻译:

汉明图和三角图上的引导渗透中的渗透集

[R图上的-Neighbor Bootstrap渗透是顶点的激活过程。该过程从一些初始激活的顶点开始,然后在每一轮中,至少包含至少一个不活动的顶点[R活动邻居被激活。导致所有顶点激活的一组初始激活顶点被称为渗滤集。表示渗滤集中的最小尺寸[R图上的邻居引导渗透过程 G 通过 G[R。在本文中,我们给出了ķñd[R,在哪里 ķñd 是的笛卡尔积 d 完整图形的副本 ķñ这就是汉明图。除其他结果外,何时d 达到无穷大,我们表明 ķñd[R=1个+Ø1个d+1个[Rd 如果 [Rd2ñ[R+1个。此外,我们明确确定大号ķñ[R,在哪里 大号ķñ 是的线图 ķñ 也称为三角图。

更新日期:2020-10-30
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