随机图上的熄灯
Lights Out On A Random Graph
摘要：
我们考虑广义游戏熄灯发挥图和调查以下问题：对于一个给定的正整数$n$，什么是概率，一个图均匀随机选择的一组图与$n$顶点产生一个普遍可解的游戏熄灯？当$n \leq 11$时，我们通过确定游戏对于每个具有$n$顶点的图是否普遍可解来精确计算这个概率。我们通过使用1，000，000次试验应用蒙特卡罗方法，用$n \leq 100$近似每个正整数$n$的概率。我们还对连通图进行了类似的计算。
英文摘要：
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.
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