# Randomly coloring simple hypergraphs with fewer colors

## TL;DR

This paper demonstrates that for simple hypergraphs with certain parameters, a Glauber Dynamics algorithm efficiently produces a near-uniform proper coloring using fewer colors than previously known, with rapid convergence.

## Contribution

It establishes improved bounds on the number of colors needed for rapid mixing of Glauber Dynamics in simple hypergraphs, advancing prior results.

## Key findings

- Glauber Dynamics mixes rapidly under new color bounds
- Efficient coloring with fewer colors in simple hypergraphs
- Improved theoretical guarantees over previous work

## Abstract

We study the problem of constructing a (near) uniform random proper $q$-coloring of a simple $k$-uniform hypergraph with $n$ vertices and maximum degree $\Delta$. (Proper in that no edge is mono-colored and simple in that two edges have maximum intersection of size one). We show that if $q\geq \max\{C_k\log n,500k^3\Delta^{1/(k-1)}\}$ then the Glauber Dynamics will become close to uniform in $O(n\log n)$ time, given a random (improper) start. This improves on the results in Frieze and Melsted [5].