{
  "id": "1402.7105",
  "version": "v1",
  "published": "2014-02-27T23:09:18.000Z",
  "updated": "2014-02-27T23:09:18.000Z",
  "title": "Fool's Solitaire on Joins and Cartesian Products of Graphs",
  "authors": [
    "Jennifer Wise",
    "Sarah Loeb"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "Peg solitaire is a game generalized to connected graphs by Beeler and Hoilman. In the game pegs are placed on all but one vertex. If $xyz$ form a 3-vertex path and $x$ and $y$ each have a peg but $z$ does not, then we can remove the pegs at $x$ and $y$ and place a peg at $z$. By analogy with the moves in the original game, this is called a jump. The goal of the peg solitaire game on graphs is to find jumps that reduce the number of pegs on the graph to 1. Beeler and Rodriguez proposed a variant where we instead want to maximize the number of pegs remaining when no more jumps can be made. Maximizing over all initial locations of a single hole, the maximum number of pegs left on a graph $G$ when no jumps remain is the fool's solitaire number $F(G)$. We determine the fool's solitaire number for the join of any graphs $G$ and $H$. For the cartesian product, we determine $F(G \\Box K_k)$ when $k \\ge 3$ and $G$ is connected and show why our argument fails when $k=2$. Finally, we give conditions on graphs $G$ and $H$ that imply $F(G \\Box H) \\ge F(G) F(H)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-27T23:09:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cartesian product",
      "fools solitaire number",
      "peg solitaire game",
      "single hole",
      "initial locations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.7105W"
    }
  }
}