We present improved bounds for randomly sampling $k$-colorings of graphs with maximum degree $Δ$; our results hold without any further structural 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 $k>2Δ$ where $Δ$ is the maximum degree of the input graph. This bound was improved by Vigoda (1999) to $k>(11/6)Δ$ using a "flip" dynamics which recolors (small) maximal two-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 $k>(11/6-\varepsilon)Δ$ where $\varepsilon\approx 10^{-5}$. 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 $Δ\geq125$ and $k\geq1.809Δ$. This yields, through recent spectral independence results, an optimal $O(n\log{n})$ mixing time for the Glauber dynamics for every fixed $Δ\geq125$ in the same range of $k/Δ$. Our proof utilizes path coupling with a simple weighted Hamming distance for "unblocked" neighbors.

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