{
  "id": "1804.03717",
  "version": "v1",
  "published": "2018-04-10T21:00:52.000Z",
  "updated": "2018-04-10T21:00:52.000Z",
  "title": "Optimal pebbling number of graphs with given minimum degree",
  "authors": [
    "Andrzej Czygrinow",
    "Glenn Hurlbert",
    "Gyula Y. Katona",
    "László F. Papp"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Consider a distribution of pebbles on a connected graph $G$. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the application of a sequence of pebbling moves. The optimal pebbling number $\\pi^*(G)$ is the smallest number of pebbles which we can distribute in such a way that each vertex is reachable. It was known that the optimal pebbling number of any connected graph is at most $\\frac{4n}{\\delta+1}$, where $\\delta$ is the minimum degree of the graph. We strengthen this bound by showing that equality cannot be attained and that the bound is sharp. If $\\operatorname{diam}(G)\\geq 3$ then we further improve the bound to $\\pi^*(G)\\leq\\frac{3.75n}{\\delta+1}$. On the other hand, we show that for arbitrary large diameter and any $\\epsilon>0$ there are infinitely many graphs whose optimal pebbling number is bigger than $\\left(\\frac{8}{3}-\\epsilon\\right)\\frac{n}{(\\delta+1)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-10T21:00:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal pebbling number",
      "minimum degree",
      "connected graph",
      "arbitrary large diameter",
      "distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}