Lights Out On A Random Graph

TL;DR

This paper investigates the probability that a randomly chosen graph with n vertices results in a universally solvable Lights Out game, providing exact probabilities for small n and approximations for larger n using Monte Carlo simulations.

Contribution

It offers the first comprehensive analysis of Lights Out solvability probabilities on random graphs, including exact calculations for small graphs and Monte Carlo estimates for larger ones.

Findings

Exact solvability probabilities for n ≤ 11

Monte Carlo approximations for n up to 100

Analysis of connected graphs' solvability probabilities

Abstract

We consider the generalized game Lights Out played on a graph and investigate the following question: for a given positive integer $n$, what is the probability that a graph chosen uniformly at random from the set of graphs with $n$ vertices yields a universally solvable game of Lights Out? When $n \leq 11$, we compute this probability exactly by determining if the game is universally solvable for each graph with $n$ vertices. We approximate this probability for each positive integer $n$ with $n \leq 100$ by applying a Monte Carlo method using 1,000,000 trials. We also perform the analogous computations for connected graphs.