Percolating sets in bootstrap percolation on the Hamming graphs
Abstract
For any integer $r\geqslant0$, 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 $m(K_n^d, r)$, where $K_n^d$ is the Cartesian product of $d$ copies of the complete graph $K_n$ which is referred as the Hamming graph. Among other results, we show that $m(K_n^d, r)=\frac{1+o(1)}{(d+1)!}r^d$ when both $r$ and $d$ go to infinity with $r<n$ and $d=o(\!\sqrt{r})$.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.01942
 Bibcode:
 2019arXiv190501942B
 Keywords:

 Mathematics  Combinatorics;
 05C35;
 60K35