{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T12:49:17Z","timestamp":1754484557365,"version":"3.41.2"},"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>The $r$-neighbour bootstrap process is an update rule for the states of vertices in which `uninfected' vertices with at least $r$ `infected' neighbours become infected and a set of initially infected vertices is said to percolate\u00a0if eventually all vertices are infected.\u00a0 For every $r \\geq 3$, a sharp condition is given for the minimum degree of a sufficiently large graph that guarantees the existence of a percolating set of size $r$.\u00a0 In the case $r=3$, for $n$ large enough, any graph on $n$ vertices with minimum degree $\\lfloor n\/2 \\rfloor +1$ has a percolating set of size $3$ and for $r \\geq 4$ and $n$ large enough (in terms of $r$), every graph on $n$ vertices with minimum degree $\\lfloor n\/2 \\rfloor + (r-3)$ has a percolating set of size $r$.\u00a0 A class of examples are given to show the sharpness of these results.<\/jats:p>","DOI":"10.37236\/6937","type":"journal-article","created":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T02:20:15Z","timestamp":1590718815000},"source":"Crossref","is-referenced-by-count":1,"title":["Minimum Degree Conditions for Small Percolating Sets in Bootstrap Percolation"],"prefix":"10.37236","volume":"27","author":[{"given":"Karen","family":"Gunderson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,5,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p37\/8094","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i2p37\/8094","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,29]],"date-time":"2020-05-29T02:20:16Z","timestamp":1590718816000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i2p37"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,29]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,4,3]]}},"URL":"https:\/\/doi.org\/10.37236\/6937","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,5,29]]},"article-number":"P2.37"}}