Fool's solitaire on joins and Cartesian products of graphs

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◻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◻H) ≥ F (G)F (H).