On Percolation and the Bunkbed Conjecture
SVANTE LINUSSON
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Source: Crossref
Published: Feb 17, 2010
DOI: 10.1017/s0963548309990666
Open original source ↗Source abstract
We study a problem on edge percolation on product graphs G × K 2 . Here G is any finite graph and K 2 consists of two vertices {0, 1} connected by an edge. Every edge in G × K 2 is present with probability p independent of other edges. The bunkbed conjecture states that for all G and p , the probability that ( u , 0) is in the same component as ( v , 0) is greater than or equal to the probability that ( u , 0) is in the same component as ( v , 1) for every pair of vertices u , v ∈ G . 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 G , in particular outerplanar graphs.
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