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- W1839626290 abstract "Graph bootstrap percolation is a simple cellular automaton introduced by Bollob'as in 1968. Given a graph $H$ and a set $G subseteq E(K_n)$ we initially all edges in $G$ and then, in consecutive steps, we infect every $e in K_n$ that completes a new infected copy of $H$ in $K_n$. We say that $G$ percolates if eventually every edge in $K_n$ is infected. The extremal question about the size of the smallest percolating sets when $H = K_r$ was answered independently by Alon, Kalai and Frankl. Here we consider a different question raised more recently by Bollob'as: what is the maximum time the process can run before it stabilizes? It is an easy observation that for $r=3$ this maximum is $lceil log_2 (n-1) rceil $. However, a new phenomenon occurs for $r=4$ when, as we show, the maximum time of the process is $n-3$. For $r geq 5$ the behaviour of the dynamics is even more complex, which we demonstrate by showing that the $K_r$-bootstrap process can run for at least $n^{2-varepsilon_r}$ time steps for some $varepsilon_r$ that tends to $0$ as $r to infty$." @default.
- W1839626290 created "2016-06-24" @default.
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- W1839626290 date "2015-10-24" @default.
- W1839626290 modified "2023-09-27" @default.
- W1839626290 title "On the maximum running time in graph bootstrap percolation" @default.
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