Lights Out On A Random Graph

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 $n\le 11$, 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 $n\le 100$ by applying a Monte Carlo method using 1,000,000 trials. We also perform the analogous computations for connected graphs.

10 pages, 1 figure. The first version incorrectly stated that the number of unlabeled graphs with n vertices was known up to n=87. It is known at least up n=140, see reference 5. Consequently, we extended our results from n=87 to n=100, which required improving our algorithm. All results in the second version were obtained using the new code