Deterministic approximate counting of colorings with fewer than $2Δ$ colors via absence of zeros

Let $Δ,q\geq 3$ be integers. We prove that there exists $η\geq 0.002$ such that if $q\geq (2-η)Δ$, then there exists an open set $\mathcal{U}\subset \mathbb{C}$ that contains the interval $[0,1]$ such that for each $w\in \mathcal{U}$ and any graph $G=(V,E)$ of maximum degree at most $Δ$, the partition function of the anti-ferromagnetic $q$-state Potts model evaluated at $w$ does not vanish. This provides a (modest) improvement on a result of Liu, Sinclair, and Srivastava, and breaks the $q=2Δ$-barrier for this problem. As a direct consequence we obtain via Barvinok’s interpolation method a deterministic polynomial time algorithm to approximate the number of proper $q$-colorings of graphs of maximum degree at most $Δ$, provided $q\geq (2-η)Δ$.

💡 Research Summary

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The paper addresses the long‑standing problem of deterministically approximating the number of proper q‑colorings of graphs with maximum degree Δ. Prior deterministic results required q ≥ 2Δ (Liu, Sinclair, and Srivastava, 2020) and relied on a zero‑free region for the anti‑ferromagnetic q‑state Potts model partition function Z_G(q,w) around the real interval