Tight Bounds for Sampling q-Colorings via Coupling from the Past

The Coupling from the Past (CFTP) paradigm is a canonical method for perfect sampling. For uniform sampling of proper $q$-colorings in graphs with maximum degree $Δ$, the bounding chains of Huber (STOC 1998) provide a systematic framework for efficiently implementing CFTP algorithms within the classical regime $q \ge (1 + o(1))Δ^2$. This was subsequently improved to $q > 3Δ$ by Bhandari and Chakraborty (STOC 2020) and to $q \ge (8/3 + o(1))Δ$ by Jain, Sah, and Sawhney (STOC 2021). In this work, we establish the asymptotically tight threshold for bounding-chain-based CFTP algorithms for graph colorings. We prove a lower bound showing that all such algorithms satisfying the standard contraction property require $q \ge 2.5Δ$, and we present an efficient CFTP algorithm that achieves this asymptotically optimal threshold $q \ge (2.5 + o(1))Δ$ via an optimal design of bounding chains.