{
  "id": "1810.05266",
  "version": "v1",
  "published": "2018-10-11T21:43:53.000Z",
  "updated": "2018-10-11T21:43:53.000Z",
  "title": "Optimal pebbling number of the square grid",
  "authors": [
    "Ervin Győri",
    "Gyula Y. Katona",
    "László F. Papp"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number $\\pi_{opt}$ is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. The optimal pebbling number of the square grid graph $P_n\\square P_m$ was investigated in several papers. In this paper, we present a new method using some recent ideas to give a lower bound on $\\pi_{opt}$. We apply this technique to prove that $\\pi_{opt}(P_n\\square P_m)\\geq \\frac{2}{13}nm$. Our method also gives a new proof for $\\pi_{opt}(P_n)=\\pi_{opt}(C_n)=\\left\\lceil\\frac{2n}{3}\\right\\rceil$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-11T21:43:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal pebbling number",
      "pebbling move",
      "pebble distribution",
      "square grid graph",
      "graph removes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}