Some Winnability Results for the Neighborhood and Group Labeling Lights Out Games

TL;DR

This paper investigates conditions under which the group labeling lights out game can be won on specific graph classes, providing new proofs and characterizations for these combinatorial puzzles.

Contribution

It establishes necessary and sufficient conditions for winning the game on paths, cycles, and complete bipartite graphs, and offers a new proof for neighborhood lights out on bipartite graphs.

Findings

Characterization of win conditions on path graphs

Characterization of win conditions on cycle graphs

New proof for neighborhood lights out on bipartite graphs

Abstract

We look at both the \emph{group labeling lights out game} and the \emph{neighborhood lights out game}. Our main focus is to determine necessary and sufficient conditions for when the group labeling lights out game on path graphs, cycle graphs, and complete bipartite graphs can be won for every possible initial labeling. In the process of solving this problem, we demonstrate a new proof for when the neighborhood lights out game on complete bipartite graphs can be won for every possible initial labeling.