- Bootstrap percolation on the Hamming graphs(arXiv)
Creator : Meysam Miralaei, Ali Mohammadian, Behruz Tayfeh-Rezaie
Summary : The r-edge bootstrap percolation on a graph is an activation technique of the sides. The method begins with some initially activated edges after which, in every spherical, any inactive edge whose one in all endpoints is incident to no less than r lively edges turns into activated. A set of initially activated edges resulting in the activation of all edges is alleged to be a percolating set. Denote the minimal dimension of a percolating set within the r-edge bootstrap percolation course of on a graph G by me(G,r). The significance of the r-edge bootstrap percolation depends on the truth that me(G,r) supplies bounds on m(G,r), that’s, the minimal dimension of a percolating set within the r-neighbor bootstrap percolation course of on G. On this paper, we explicitly decide me(Kdn,r), the place Kdn is the Cartesian product of d copies of the whole graph on n vertices which is referred as Hamming graph. Utilizing this, we present that m(Kdn,r)=(1+o(1))dr−1r! when n,r are mounted and d goes to infinity which extends a recognized outcome on hypercubes
