- Bootstrap percolation on the Hamming graphs(arXiv)
Creator : Meysam Miralaei, Ali Mohammadian, Behruz Tayfeh-Rezaie
Abstract : The r-edge bootstrap percolation on a graph is an activation strategy of the edges. The tactic begins with some initially activated edges after which, in each spherical, any inactive edge whose one amongst endpoints is incident to a minimum of r energetic edges turns into activated. A set of initially activated edges ensuing within the activation of all edges is alleged to be a percolating set. Denote the minimal dimension of a percolating set inside the r-edge bootstrap percolation course of on a graph G by me(G,r). The importance of the r-edge bootstrap percolation relies on the reality that me(G,r) provides bounds on m(G,r), that is, the minimal dimension of a percolating set inside the r-neighbor bootstrap percolation course of on G. On this paper, we explicitly determine me(Kdn,r), the place Kdn is the Cartesian product of d copies of the entire graph on n vertices which is referred as Hamming graph. Using this, we current that m(Kdn,r)=(1+o(1))dr−1r! when n,r are mounted and d goes to infinity which extends a acknowledged end result on hypercubes
