Randomly colouring simple hypergraphs

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 Δ. (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,Δso 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).