Fool’s solitaire on joins and Cartesian products of graphs

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 has 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).