Sampling Colorings with Fixed Color Class Sizes

2026-03-09

AI Summary

This study addresses the problem of efficiently sampling q-colorings with balanced color class sizes—either exactly or approximately fixed—in graphs of maximum degree Δ, where the number of colors satisfies q > 2Δ. We present the first polynomial-time sampling algorithm for this task, grounded in a multivariate polynomial geometric framework and integrating probabilistic analysis with combinatorial optimization techniques. Our main contributions include establishing the existence of such balanced colorings, proving a multivariate local central limit theorem that characterizes the distribution of color class sizes, and extending the algorithm to accommodate small deviations from perfect balance. These results collectively reveal the underlying statistical structure of balanced colorings and enable efficient approximate uniform sampling under the given constraints.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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.

Problem

• equitable coloring
• random sampling
• color class sizes
• graph coloring
• uniform sampling

Innovation

• equitable coloring
• polynomial-time sampling
• geometry of polynomials
• local central limit theorem
• color class sizes