The time of graph bootstrap percolation. Issue 1 (26th August 2016)

• Record Type: Journal Article
• Authors: Gunderson, Karen; Koch, Sebastian; Przykucki, Michał
• Is Part Of: Random structures & algorithms. Volume 51:Issue 1(2017)
• Page Start: 143
• Page End: 168
• Publication Date: 2016-08-26
• Subjects: bootstrap percolation -- weak saturation -- random graphs; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20660
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: Graph bootstrap percolation, introduced by Bollobás in 1968, is a cellular automaton defined as follows. Given a "small" graph H and a "large" graph G = G 0 ⊆ K n, in consecutive steps we obtain G t + 1 from G t by adding to it all new edges e such that G t ∪ e contains a new copy of H . We say that G percolates if for some t ≥ 0, we have G t = K n . For H = K r, the question about the size of the smallest percolating graphs was independently answered by Alon, Frankl and Kalai in the 1980's. Recently, Balogh, Bollobás and Morris considered graph bootstrap percolation for G = G ( n, p ) and studied the critical probability p c ( n, K r ), for the event that the graph percolates with high probability. In this paper, using the same setup, we determine, up to a logarithmic factor, the critical probability for percolation by time t for all 1 ≤ t ≤ C log log n © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 143–168, 2017