Lights Out on finite graphs

Abstract

Lights Out is a one-player game played on a finite graph. In the standard game the vertices can be either on or off; pressing a vertex toggles its state and that of all adjacent vertices. The goal of the game is to turn off all of the lights. We study an extension of the game in which the state of a vertex may be one of a finite number of colors. We determine which graphs in certain families (spider graphs and generalized theta graphs) are winnable for every initial coloring. We also provide a construction that gives every always-winnable tree for any prime power number of colors.