Graph Powers of Groups

TL;DR

This paper generalizes the Lights Out puzzle to group states on graphs, analyzing the structure of achievable configurations and identifying conditions under which the problem simplifies to linear algebra.

Contribution

It introduces a group-theoretic framework for the Lights Out puzzle, extends analysis to non-abelian groups, and characterizes RA graphs using lattice spanning conditions.

Findings

Most graphs are RA, simplifying analysis to linear algebra.

Odd-dimensional cubes and folded cubes are not RA.

A lattice spanning condition characterizes RA graphs with Heisenberg groups.

Abstract

The Lights Out Puzzle, played on a graph $\Gamma$, has been studied using linear algebra over $\mathbb{F}_2$ and more generally over $\mathbb{Z}/k\mathbb{Z}$. We generalize the setting by allowing the states of vertices to be the elements of a group $G$, where a \textit{click} in vertex $v$ multiplies the state of $v$ and its neighbors by an element $g \in G$ on the right. Starting with the identity element $e \in G$ for all vertices, the totality of all achievable state configurations forms a group $G^{\Gamma}$. This group generalizes parallel products of group actions and provides a rich structure for analysis. For many graphs, which we term ``RA'' (reducible to abelian), the problem reduces -- regardless of $G$ -- to a linear algebra question over $\mathbb{Z}$. We discuss a chain of five different subgroups consisting of commutators and introduce techniques for showing that families of graphs are RA using each. In particular, using Heisenberg groups, we establish that a graph is RA precisely when a certain lattice spans $\mathbb{Z}^{|\Gamma|}$. While most graphs appear to be RA, we show the odd-dimensional cube graphs $Q_{2n+1}$ and folded cube graphs $\square_d$, for $d$ odd or 2, are not.