{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T13:45:01Z","timestamp":1768743901293,"version":"3.49.0"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2010,2,17]],"date-time":"2010-02-17T00:00:00Z","timestamp":1266364800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>We study a problem on edge percolation on product graphs<jats:italic>G<\/jats:italic>\u00d7<jats:italic>K<\/jats:italic><jats:sub>2<\/jats:sub>. Here<jats:italic>G<\/jats:italic>is any finite graph and<jats:italic>K<\/jats:italic><jats:sub>2<\/jats:sub>consists of two vertices {0, 1} connected by an edge. Every edge in<jats:italic>G<\/jats:italic>\u00d7<jats:italic>K<\/jats:italic><jats:sub>2<\/jats:sub>is present with probability<jats:italic>p<\/jats:italic>independent of other edges. The bunkbed conjecture states that for all<jats:italic>G<\/jats:italic>and<jats:italic>p<\/jats:italic>, the probability that (<jats:italic>u<\/jats:italic>, 0) is in the same component as (<jats:italic>v<\/jats:italic>, 0) is greater than or equal to the probability that (<jats:italic>u<\/jats:italic>, 0) is in the same component as (<jats:italic>v<\/jats:italic>, 1) for every pair of vertices<jats:italic>u<\/jats:italic>,<jats:italic>v<\/jats:italic>\u2208<jats:italic>G<\/jats:italic>.<\/jats:p><jats:p>We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs<jats:italic>G<\/jats:italic>, in particular outerplanar graphs.<\/jats:p>","DOI":"10.1017\/s0963548309990666","type":"journal-article","created":{"date-parts":[[2010,2,17]],"date-time":"2010-02-17T11:58:48Z","timestamp":1266407928000},"page":"103-117","source":"Crossref","is-referenced-by-count":7,"title":["On Percolation and the Bunkbed Conjecture"],"prefix":"10.1017","volume":"20","author":[{"given":"SVANTE","family":"LINUSSON","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2010,2,17]]},"reference":[{"key":"S0963548309990666_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03981-6"},{"key":"S0963548309990666_ref12","unstructured":"[12] Linusson S. (2009) A note on correlations in randomly oriented graphs. Preprint. arXiv:0905.2881"},{"key":"S0963548309990666_ref2","unstructured":"[2] Alm S. E. and Linusson S. (2009) Correlations for paths in random orientations of G(n, p). Preprint. arXiv:0906.0720"},{"key":"S0963548309990666_ref13","doi-asserted-by":"publisher","DOI":"10.2307\/1426466"},{"key":"S0963548309990666_ref1","unstructured":"[1] Alm S. E. and Linusson S. (2009) A counter-intuitive correlation in a random tournament. Combin. Probab. Comput., to appear. arXiv:0906.0240"},{"key":"S0963548309990666_ref11","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240010106"},{"key":"S0963548309990666_ref9","volume-title":"Proc. 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