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 $\mathrm{\Delta }$, the bounding chains of Huber (STOC 1998) provide a systematic framework for efficiently implementing CFTP algorithms within the classical regime $q\ge (1+o(1)){\mathrm{\Delta }}^2$. This was subsequently improved to $q>3\mathrm{\Delta }$ by Bhandari and Chakraborty (STOC 2020) and to $q\ge (8\mathrm{/}3+o(1))\mathrm{\Delta }$ 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 $q\ge 2.5\mathrm{\Delta }$, and we present an efficient CFTP algorithm that achieves this asymptotically optimal threshold $q\ge (2.5+o(1))\mathrm{\Delta }$ via an optimal design of bounding chains.