We present improved bounds for randomly sampling $k$-colorings of graphs with maximum degree $\Delta$; our results hold without any further assumptions on the graph. The Glauber dynamics is a simple single-site update Markov chain. Jerrum (1995) proved an optimal $O(n\log{n})$ mixing time bound for Glauber dynamics whenever $k>2\Delta$ where $\Delta$ is the maximum degree of the input graph. This bound was improved by Vigoda (1999) to $k > (11/6)\Delta$ using a "flip" dynamics which recolors (small) maximal 2-colored components in each step. Vigoda's result was the best known for general graphs for 20 years until Chen et al. (2019) established optimal mixing of the flip dynamics for $k > (11/6 - \epsilon ) \Delta$ where $\epsilon \approx 10^{-5}$. We present the first substantial improvement over these results. We prove an optimal mixing time bound of $O(n\log{n})$ for the flip dynamics when $k \geq 1.809 \Delta$. This yields, through recent spectral independence results, an optimal $O(n\log{n})$ mixing time for the Glauber dynamics for the same range of $k/\Delta$ when $\Delta=O(1)$. Our proof utilizes path coupling with a simple weighted Hamming distance for "unblocked" neighbors.

翻译：我们针对最大度为$\Delta$的图的$k$-着色随机采样问题提出了改进的界限；我们的结果无需对图作任何额外假设。格劳伯动力学是一种简单的单点更新马尔可夫链。Jerrum（1995）证明了当$k>2\Delta$（其中$\Delta$为输入图的最大度）时，格劳伯动力学的混合时间最优界为$O(n\log{n})$。Vigoda（1999）通过采用在每一步对（小型）最大双色连通分量进行重着色的"翻转"动力学，将此界限改进至$k > (11/6)\Delta$。在随后20年间，Vigoda的结果一直是普通图的最佳已知界限，直至Chen等人（2019）针对$k > (11/6 - \epsilon ) \Delta$（其中$\epsilon \approx 10^{-5}$）建立了翻转动力学的最优混合性。我们首次对这些结果作出实质性改进：当$k \geq 1.809 \Delta$时，我们证明了翻转动力学具有$O(n\log{n})$的最优混合时间界。结合近期谱独立性研究，当$\Delta=O(1)$时，该结果可为相同$k/\Delta$范围内的格劳伯动力学导出最优的$O(n\log{n})$混合时间。我们的证明采用路径耦合方法，并为"非阻塞"邻域配置了简单的加权汉明距离度量。