Sampling Colorings with Fixed Color Class Sizes

TL;DR

This paper develops a polynomial-time algorithm for approximately sampling equitable graph colorings with a number of colors greater than twice the maximum degree, extending sampling methods to constrained colorings.

Contribution

It introduces a novel polynomial-time sampling algorithm for equitable colorings when the number of colors exceeds twice the maximum degree, expanding sampling techniques to constrained colorings.

Findings

Provides a polynomial-time sampling algorithm for equitable colorings with q > 2Δ.

Extends sampling methods to colorings with small deviations from equitable.

Establishes a multivariate local Central Limit Theorem for color class sizes.

Abstract

In 1970 Hajnal and Szemer\'edi proved a conjecture of Erd\"os that for a graph with maximum degree $\Delta$, there exists an equitable $\Delta+1$ 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 $q\geq 1.809 \Delta$. In this paper we give a polynomial-time sampling algorithm for equitable colorings when $q> 2\Delta$. 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.