Sampling Colorings Close to the Maximum Degree: Non-Markovian Coupling and Local Uniformity

Sampling graph colorings via local Markov chains is a central problem in approximate counting and Markov chain Monte Carlo (MCMC). We address the problem of sampling a random $k$-coloring of a graph with maximum degree $Δ$. The simplest algorithmic approach is to establish rapid mixing of the single-site update chain known as the Metropolis Glauber dynamics, which at each step chooses a random vertex $v$ and proposes a random color $c$, recoloring $v$ to $c$ if the resulting coloring remains proper. It is a long-standing open problem to prove that the Glauber dynamics has polynomial mixing time on all graphs whenever $k\geqΔ+2$. We prove that for every $δ>0$ and all $Δ\geq Δ_0(δ)$, if $k\ge (1+δ)Δ$ then the Glauber dynamics has optimal mixing time of $O_δ(|V| \log |V|)$ on any graph of girth $\geq 7$ and maximum degree $Δ$. Our approach builds on a non-Markovian coupling introduced by Hayes and Vigoda (2003) for the large-degree regime $Δ=Ω(\log n)$ and girth $11$, in which updates at time $t$ may depend on and modify proposed updates at future times. A complete analysis of this framework requires resolving substantial technical obstacles that remain in the original argument, and extending it to the constant-degree regime introduces further difficulties, since non-Markovian updates may fail with constant probability. We overcome these obstacles by developing and analyzing a refined local non-Markovian coupling, and by establishing new local-uniformity results for the Metropolis dynamics, extending prior results for the heat-bath chain due to Hayes (2013). Together, these ingredients provide a complete analysis of the non-Markovian coupling framework in the large-degree regime, while simultaneously strengthening it substantially to obtain optimal mixing all the way down to the constant-degree setting.