Randomly coloring simple hypergraphs

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 ∆. (Proper in that no edge is mono-colored and simple in that two edges have maximum intersection of size one). We show that if for some α < 1 we have ∆ ≥ n α and q ≥ ∆(1+α)/kα then Glauber dynamics will become close to uniform in O(n log n) time from a random (improper) start. Note that for k > 1 + α −1 we can take q = o(∆).