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We present a randomized algorithm which takes as input an undirected graph$G$ on $n$ vertices with maximum degree $\Delta$, and a number of colors $k\geq (8/3 + o_{\Delta}(1))\Delta$, and returns -- in expected time$\tilde{O}(n\Delta^{2}\log{k})$ -- a proper $k$-coloring of $G$ distributedperfectly uniformly on the set of all proper $k$-colorings of $G$. Notably, oursampler breaks the barrier at $k = 3\Delta$ encountered in recent work ofBhandari and Chakraborty [STOC 2020]. We also sketch how to modify our methodsto relax the restriction on $k$ to $k \geq (8/3 - \epsilon_0)\Delta$ for anabsolute constant $\epsilon_0 > 0$.  As in the work of Bhandari and Chakraborty, and the pioneering work of Huber[STOC 1998], our sampler is based on Coupling from the Past [Propp&Wilson,Random Struct. Algorithms, 1995] and the bounding chain method [Huber, STOC1998; H\"aggstr\"om&Nelander, Scand. J. Statist., 1999]. Our innovationsinclude a novel bounding chain routine inspired by Jerrum's analysis of theGlauber dynamics [Random Struct. Algorithms, 1995], as well as apreconditioning routine for bounding chains which uses the algorithmic Lov\'aszLocal Lemma [Moser&Tardos, J.ACM, 2010].

Perfectly Sampling $k\geq (8/3 +o(1))\Delta$-Colorings in Graphs