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

TL;DR

This paper proves the absence of zeros in the partition function of the anti-ferromagnetic Potts model for certain parameters, leading to a deterministic polynomial-time algorithm for counting proper colorings in graphs with bounded degree.

Contribution

It extends the range of parameters for which the partition function is non-vanishing, enabling efficient counting algorithms beyond the previous $2\Delta$-colors barrier.

Findings

Established a non-vanishing region for the partition function for $q extless (2- ext{ extless} ext{eta})\Delta$

Developed a deterministic polynomial-time approximation algorithm for counting proper colorings

Improved previous bounds on the color count threshold for efficient counting

Abstract

Let $\Delta,q\geq 3$ be integers. We prove that there exists $\eta\geq 0.002$ such that if $q\geq (2-\eta)\Delta$, 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 $\Delta$, 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 $q=2\Delta$-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 $\Delta$, provided $q\geq (2-\eta)\Delta$.