Lights Out On Nearly Complete Graphs
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908487591657472 |
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| author | Forrest, Bradley Goyal, Riya |
| author_facet | Forrest, Bradley Goyal, Riya |
| contents | 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 $\binom{n}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09341 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lights Out On Nearly Complete Graphs Forrest, Bradley Goyal, Riya Combinatorics 05C57 (Primary) 05C80, 60C99 (Secondary) 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 $\binom{n}{2}$. |
| title | Lights Out On Nearly Complete Graphs |
| topic | Combinatorics 05C57 (Primary) 05C80, 60C99 (Secondary) |
| url | https://arxiv.org/abs/2508.09341 |