{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T21:06:35Z","timestamp":1775163995064,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Graph bootstrap percolation is a simple cellular automaton introduced by Bollob\u00e1s in\u00a01968. Given a graph $H$ and a set $G \\subseteq E(K_n)$ we initially `infect' all edges\u00a0in $G$ and then, in consecutive steps, we infect every $e \\in K_n$ that completes a new\u00a0infected copy of $H$ in $K_n$. We say that $G$ percolates if eventually every edge in\u00a0$K_n$ is infected. The extremal question about the size of the smallest percolating sets\u00a0when $H = K_r$ was answered independently by Alon, Kalai and Frankl.\u00a0Here we consider a different question raised more recently by Bollob\u00e1s: what is\u00a0the maximum time the process can run before it stabilizes?\u00a0It is an easy observation that\u00a0for $r=3$ this maximum is $\\lceil \\log_2 (n-1) \\rceil $. However, a new phenomenon occurs\u00a0for $r=4$ when, as we show, the maximum time of the process is $n-3$. For $r \\geq 5$ the\u00a0behaviour of the dynamics is even more complex, which we demonstrate by showing that the\u00a0$K_r$-bootstrap process can run for at least $n^{2-\\varepsilon_r}$ time steps for some\u00a0$\\varepsilon_r$ that tends to $0$ as $r \\to \\infty$.<\/jats:p>","DOI":"10.37236\/5771","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:55:43Z","timestamp":1578671743000},"source":"Crossref","is-referenced-by-count":6,"title":["On the Maximum Running Time in Graph Bootstrap Percolation"],"prefix":"10.37236","volume":"24","author":[{"given":"B\u00e9la","family":"Bollob\u00e1s","sequence":"first","affiliation":[]},{"given":"Micha\u0142","family":"Przykucki","sequence":"additional","affiliation":[]},{"given":"Oliver","family":"Riordan","sequence":"additional","affiliation":[]},{"given":"Julian","family":"Sahasrabudhe","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,5,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p16\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p16\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:57:27Z","timestamp":1579237047000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i2p16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,5]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/5771","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,5,5]]},"article-number":"P2.16"}}