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- W4288283962 abstract "Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollob'as in 1968. In this process, we start with initial infected set of edges $E_0$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E_tsubseteq E(K_n)$, we infect a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n],E_tcup e)$. A question raised by Bollob'as asks for the maximum time the process can run before it stabilizes. Bollob'as, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H=K_r$. They answered the question for $rleq 4$ and gave a non-trivial lower bound for every $rgeq 5$. They also conjectured that the maximal running time is $o(n^2)$ for every integer $r$. In this paper we disprove their conjecture for every $rgeq 6$ and we give a better lower bound for the case $r=5$; in the proof we use the Behrend construction." @default.
- W4288283962 created "2022-07-28" @default.
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- W4288283962 date "2019-07-10" @default.
- W4288283962 modified "2023-09-28" @default.
- W4288283962 title "The maximum length of $K_r$-Bootstrap Percolation" @default.
- W4288283962 doi "https://doi.org/10.48550/arxiv.1907.04559" @default.
- W4288283962 hasPublicationYear "2019" @default.
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