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

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.