It is well known that for every even integer n, the complete graph $$K_{n}$$ has a one-factorization, namely a proper edge coloring with $$n-1$$ colors. Unfortunately, not much is known about the possible structure of large one-factorizations. Also, at present we have only woefully few explicit constructions of large one-factorizations. In particular, we know essentially nothing about the typical properties of one-factorizations for large n. In this view, it is desirable to find rapidly mixing Markov chains that generate one-factorizations uniformly and efficiently. No such Markov chain is currently known, and here we take a step this direction. We construct a Markov chain whose states are all edge colorings of $$K_n$$ by $$n-1$$ colors. This chain is invariant under arbitrary renaming of the vertices or the colors. It reaches a one factorization in polynomial(n) steps from every starting state. In addition, at every transition only O(n) edges change their color. We also raise some related questions and conjectures, and present results of numerical simulations of simpler variants of this Markov chain.

中文翻译：

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生成单因式分解的高效、局部和对称马尔可夫链

众所周知，对于每一个偶数 n，完全图 $$K_{n}$$ 都有一个单因式分解，即具有 $$n-1$$ 颜色的适当边着色。不幸的是，关于大型单因式分解的可能结构知之甚少。此外，目前我们只有很少的大型单因式分解的显式构造。特别是，我们对大 n 的单因式分解的典型性质基本上一无所知。在这种观点下，需要找到快速混合的马尔可夫链，以均匀有效地生成单因式分解。目前还没有这样的马尔可夫链，我们在这里朝这个方向迈出了一步。我们构造了一个马尔可夫链，其状态都是 $$K_n$$ 乘 $$n-1$$ 种颜色的边着色。该链在顶点或颜色的任意重命名下是不变的。它从每个起始状态在 polynomial(n) 步骤中达到一个分解。此外，在每次转换时，只有 O(n) 条边会改变它们的颜色。我们还提出了一些相关的问题和猜想，并展示了该马尔可夫链的更简单变体的数值模拟结果。