In this paper, we exhibit an irreducible Markov chain $M$ on the edge $k$-colorings of bipartite graphs based on certain properties of the solution space. We show that diameter of this Markov chain grows linearly with the number of edges in the graph. We also prove a polynomial upper bound on the inverse of acceptance ratio of the Metropolis-Hastings algorithm when the algorithm is applied on $M$ with the uniform distribution of all possible edge $k$-colorings of $G$. A special case of our results is the solution space of the possible completions of Latin rectangles.

中文翻译：

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二部图边缘着色的解空间上的马尔可夫链

在本文中，我们基于解空间的某些属性，在二部图的边缘$ k $-着色上展示了不可约的马尔可夫链$ M $。我们表明，该马尔可夫链的直径随着图中​​边数的增加而线性增长。我们还证明了将Metropolis-Hastings算法应用于$ M $时，Metropolis-Hastings算法的接受率倒数的多项式上限，并且所有可能的边缘$ k $-颜色$ G $都均匀分布。我们的结果的一个特例是拉丁矩形可能补全的解空间。