{
  "id": "1406.5147",
  "version": "v1",
  "published": "2014-06-19T18:47:38.000Z",
  "updated": "2014-06-19T18:47:38.000Z",
  "title": "Sandpiles, spanning trees, and plane duality",
  "authors": [
    "Melody Chan",
    "Darren Glass",
    "Matthew Macauley",
    "David Perkinson",
    "Caryn Werner",
    "Qiaoyu Yang"
  ],
  "comment": "13 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well-known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-19T18:47:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E18",
      "05C05",
      "05C25"
    ],
    "keywords": [
      "spanning trees",
      "plane duality",
      "sandpile group",
      "rotor-routing action",
      "root vertex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.5147C"
    }
  }
}