Percolating sets in bootstrap percolation on the Hamming graphs and triangular graphs

Abstract

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 r≫d2 and n⩾r+1. Furthermore, we explicitly determine m(L(Kn),r), where L(Kn) is the line graph of Kn also known as triangular graph.