{
  "id": "0811.0949",
  "version": "v2",
  "published": "2008-11-06T15:38:15.000Z",
  "updated": "2009-11-30T10:31:00.000Z",
  "title": "On percolation and the bunkbed conjecture",
  "authors": [
    "Svante Linusson"
  ],
  "comment": "13 pages, improved exposition thanks to anonymous referee. To appear in CPC",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We study a problem on edge percolation on product graphs $G\\times K_2$. Here $G$ is any finite graph and $K_2$ consists of two vertices $\\{0,1\\}$ connected by an edge. Every edge in $G\\times K_2$ 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 $u,v\\in G$. 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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-11-30T10:31:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60C05",
      "60K35"
    ],
    "keywords": [
      "probability",
      "edge percolation",
      "similar statements",
      "product graphs",
      "finite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.0949L"
    }
  }
}