An Extremal Problem for the Neighborhood Lights Out Game

TL;DR

This paper investigates the maximum number of edges in graphs where the Neighborhood Lights Out game is always winnable, introducing new game variants and extremal graph characterizations.

Contribution

It characterizes extremal graphs with nearly complete edge counts and minimum degree conditions for the Neighborhood Lights Out game, introducing a new matrix-based game variant.

Findings

Characterized extremal graphs with inom{n}{2} - c edges for c ≤ ceil n/2 ceil + 3.

Identified graphs with minimum degree n-3 where the game is always winnable.

Developed a new matrix-based version of the Lights Out game.

Abstract

Neighborhood Lights Out is a game played on graphs. Begin with a graph and a vertex labeling of the graph from the set $\{0,1,2,\dots, \ell-1\}$ for $\ell \in \mathbb{N}$. The game is played by toggling vertices: when a vertex is toggled, that vertex and each of its neighbors has its label increased by $1$ (modulo $\ell$). The game is won when every vertex has label 0. For any $n\in\mathbb{N}$ it is clear that one cannot win the game on $K_n$ unless the initial labeling assigns all vertices the same label. Given that the $K_n$ has the maximum number of edges of any simple graph on $n$ vertices it is natural to ask how many edges can be in a graph so that the Neighborhood Lights Out game is winnable regardless of the initial labeling. We find all such extremal graphs on $n$ vertices that have $\binom{n}{2} - c$ edges for $c\leq \lceil\frac{n}{2}\rceil +3$ and all those that have minimum degree $n-3$. The proofs of our results require us to introduce a new version of the Lights Out game that can be played given any square matrix.