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- W2091222603 abstract "Graph bootstrap percolation is a deterministic cellular automaton which was introduced by Bollobás in 1968, and is defined as follows. Given a graph H, and a set documentclass{article}usepackage{mathrsfs}usepackage{amsmath, amssymb}pagestyle{empty}begin{document}begin{align*}G subset E(K_n)end{align*} end{document} of initially ‘infected’ edges, we infect, at each time step, a new edge e if there is a copy of H in documentclass{article}usepackage{mathrsfs}usepackage{amsmath, amssymb}pagestyle{empty}begin{document}begin{align*}K_nend{align*} end{document} such that e is the only not-yet infected edge of H. We say that G percolates in the H-bootstrap process if eventually every edge of documentclass{article}usepackage{mathrsfs}usepackage{amsmath, amssymb}pagestyle{empty}begin{document}begin{align*}K_nend{align*} end{document} is infected. The extremal questions for this model, when H is the complete graph documentclass{article}usepackage{mathrsfs}usepackage{amsmath, amssymb}pagestyle{empty}begin{document}begin{align*}K_rend{align*} end{document}, were solved (independently) by Alon, Kalai and Frankl almost thirty years ago. In this paper we study the random questions, and determine the critical probability documentclass{article}usepackage{mathrsfs}usepackage{amsmath, amssymb}pagestyle{empty}begin{document}begin{align*}p_c(n,K_r)end{align*} end{document} for the documentclass{article}usepackage{mathrsfs}usepackage{amsmath, amssymb}pagestyle{empty}begin{document}begin{align*}K_rend{align*} end{document}-process up to a poly-logarithmic factor. In the case documentclass{article}usepackage{mathrsfs}usepackage{amsmath, amssymb}pagestyle{empty}begin{document}begin{align*}r = 4end{align*} end{document} we prove a stronger result, and determine the threshold for documentclass{article}usepackage{mathrsfs}usepackage{amsmath, amssymb}pagestyle{empty}begin{document}begin{align*}p_c(n,K_4)end{align*} end{document}. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012" @default.
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- W2091222603 date "2012-09-04" @default.
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- W2091222603 title "Graph bootstrap percolation" @default.
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- W2091222603 doi "https://doi.org/10.1002/rsa.20458" @default.
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