{
  "id": "1006.3049",
  "version": "v2",
  "published": "2010-06-15T18:40:15.000Z",
  "updated": "2015-03-20T10:51:01.000Z",
  "title": "Long paths and cycles in subgraphs of the cube",
  "authors": [
    "Eoin Long"
  ],
  "comment": "27 pages, 6 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $Q_n$ denote the graph of the $n$-dimensional cube with vertex set $\\{0,1\\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. Suppose $G$ is a subgraph of $Q_n$ with average degree at least $d$. How long a path can we guarantee to find in $G$? Our aim in this paper is to show that $G$ must contain an exponentially long path. In fact, we show that if $G$ has minimum degree at least $d$ then $G$ must contain a path of length $2^d-1$. Note that this bound is tight, as shown by a $d$-dimensional subcube of $Q_n$. We also obtain the slightly stronger result that $G$ must contain a cycle of length at least $2^d$ and prove analogous results for other `product-type' graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-06-15T18:40:15.000Z",
      "abstract": "Let $Q_n$ denote the graph of the $n$-dimensional cube with vertex set $\\{0,1\\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. Suppose $G$ is a subgraph of $Q_n$ with average degree at least $d$. How long a path can we guarantee to find in $G$? Our aim in this paper is to show that $G$ must contain an exponentially long path. In fact, we show that if $G$ has minimum degree at least $d$ then $G$ must contain a path of length $2^d-1$. Note that this bound is tight, as shown by a $d$-dimensional subcube of $Q_n$. We also obtain the slightly stronger result that $G$ must contain a cycle of length at least $2^d$.",
      "comment": "24 pages, 6 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-20T10:51:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C38"
    ],
    "keywords": [
      "vertex set",
      "average degree",
      "dimensional cube",
      "exponentially long path",
      "minimum degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.3049L"
    }
  }
}