Lights Out On Nearly Complete Graphs

We study the generalization of the game Lights Out in which the standard square grid board is replaced by a graph. We examine the probability that, when a graph is chosen uniformly at random from the set of graphs with $n$ vertices and $e$ edges, the resulting game of Lights Out is universally solvable. Our work focuses on nearly complete graphs, graphs for which $e$ is close to $\binom{n}{2}$. For large values of $n$, we prove that, among nearly complete graphs, the probability of selecting a graph that gives a universally solvable game of Lights Out is maximized when $e = \binom{n}{2} - \lfloor \frac{n}{2} \rfloor$. More specifically, we prove that for any fixed integer $m > 0$, as $n$ approaches $\infty$, this value of $e$ maximizes the probability over all values of $e$ from $\binom{n}{2} - \lfloor \frac{n}{2} \rfloor - m$ to $\binom{n}{2}$.