{
  "id": "1009.5162",
  "version": "v1",
  "published": "2010-09-27T05:36:01.000Z",
  "updated": "2010-09-27T05:36:01.000Z",
  "title": "Bounds on the Rubbling and Optimal Rubbling Numbers of Graphs",
  "authors": [
    "Gyula Y. Katona",
    "Nandor Sieben"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices $v$ and $w$ adjacent to a vertex $u$, and an extra pebble is added at vertex $u$. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number is the smallest number $m$ needed to guarantee that any vertex is reachable from any pebble distribution of $m$ pebbles. The optimal rubbling number is the smallest number $m$ needed to guarantee a pebble distribution of $m$ pebbles from which any vertex is reachable. We give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of $n$-vertex, diameter $d$ graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-27T05:36:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C99"
    ],
    "keywords": [
      "optimal rubbling number",
      "pebble distribution",
      "smallest number",
      "sharp upper bound",
      "additional move"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.5162K"
    }
  }
}