Resolution of Conjectures Related to Lights Out! and Cartesian Products

TL;DR

This paper resolves two conjectures related to the Lights Out! game played on Cartesian product graphs, advancing understanding of the game's mathematical structure and solution strategies.

Contribution

It provides proofs for two conjectures concerning Lights Out! on Cartesian product graphs, clarifying the game's underlying algebraic properties.

Findings

Confirmed conjectures about Lights Out! on Cartesian products

Enhanced understanding of the game's algebraic structure

Potential implications for solving Lights Out! configurations

Abstract

Lights Out! is a game played on a $5 \times 5$ grid of lights, or more generally on a graph. Pressing lights on the grid allows the player to turn off neighboring lights. The goal of the game is to start with a given initial configuration of lit lights and reach a state where all lights are out. Two conjectures posed in a recently published paper about Lights Out! on Cartesian products of graphs are resolved.