2014/02/27 by Jennifer Wise, Wise, Jennifer, Sarah Loeb +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Artificial Intelligence in Games #Graph Labeling and Dimension Problems #cs.DM #math.CO
paper · pdf · doi:10.48550/arxiv.1402.7105
arxiv created 2014/02/27 · arxiv updated 2014/03/03
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 Kk) when k ≥ 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) ≥ F(G) F(H).