Sampling Colorings with Fixed Color Class Sizes

TL;DR

A polynomial-time sampling algorithm for equitable colorings when $q>2\Delta$ is given, which establishes a multivariate local Central Limit Theorem for color class sizes of uniform random colorings.

Abstract

In 1970 Hajnal and Szemerédi proved a conjecture of Erdös that for a graph with maximum degree $Δ$, there exists an equitable $Δ+1$ coloring; that is a coloring where color class sizes differ by at most $1$. In 2007 Kierstand and Kostochka reproved their result and provided a polynomial-time algorithm which produces such a coloring. In this paper we study the problem of approximately sampling uniformly random equitable colorings. A series of works gives polynomial-time sampling algorithms for colorings without the color class constraint, the latest improvement being by Carlson and Vigoda for $q\geq 1.809 Δ$. In this paper we give a polynomial-time sampling algorithm for equitable colorings when $q> 2Δ$. Moreover, our results extend to colorings with small deviations from equitable (and as a corollary, establishing their existence). The proof uses the framework of the geometry of polynomials for multivariate polynomials, and as a consequence establishes a multivariate local Central Limit Theorem for color class sizes of uniform random colorings.

Figures (1)

• Figure 1: Proof outline

Theorems & Definitions (128)

• Conjecture 1
• Definition 1: $\vec{n}$-coloring
• Definition 2: Equitable colorings
• Definition 3
• Theorem 1.1: Sampling Equitable Colorings
• Theorem 1.2: Sampling Skewed Colorings
• Corollary 1
• Conjecture 2
• Conjecture 3
• Definition 4: Anti-ferromagnetic Potts model