Lights Out On Nearly Complete Graphs

TL;DR

This paper investigates the probability of universal solvability in a generalized Lights Out game on nearly complete graphs, proving that the highest probability occurs at a specific edge count close to the complete graph.

Contribution

It establishes the exact edge count in nearly complete graphs that maximizes the probability of universal solvability for large graphs.

Findings

Maximum probability at e = C(n,2) - floor(n/2)

Probability increases as e approaches this value from below

Results hold asymptotically as n approaches infinity

Abstract

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 $\binom{n}{2}$.