Fool's Solitaire on Joins and Cartesian Products of Graphs

TL;DR

This paper investigates the fool's solitaire number on graph joins and Cartesian products, providing formulas and conditions for various graph classes to understand the maximum pegs remaining in a peg solitaire game generalized to graphs.

Contribution

It determines the fool's solitaire number for joins of graphs and for Cartesian products with complete graphs, and establishes conditions for the product's fool's solitaire number to be at least the product of individual numbers.

Findings

F(G ⨝ H) is determined for graph joins.

F(G □ K_k) is found for connected G when k ≥ 3.

Conditions are given for F(G □ H) ≥ F(G)F(H).

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 $F(G \Box H) \ge F(G) F(H)$.