Randomly colouring simple hypergraphs

TL;DR

This paper investigates conditions under which Glauber dynamics efficiently produces proper colorings of simple hypergraphs, especially when the number of colors is small relative to maximum degree, advancing understanding of hypergraph coloring algorithms.

Contribution

It establishes new bounds on the number of colors needed for rapid convergence of Glauber dynamics in hypergraph coloring, particularly for k ≥ 3.

Findings

Glauber dynamics converges in O(n log n) time under certain conditions.

For k ≥ 3, the number of colors q can be smaller than the maximum degree Δ.

Conditions relate q and Δ to ensure proper coloring with high probability.

Abstract

We study the problem of constructing a (near) random proper $q$-colouring of a simple k-uniform hypergraph with n vertices and maximum degree \Delta. (Proper in that no edge is mono-coloured and simple in that two edges have maximum intersection of size one). We give conditions on q,\Delta so that if these conditions are satisfied, Glauber dynamics will converge in O(n\log n) time from a random (improper) start. The interesting thing here is that for k\geq 3 we can take q=o(\D).