Randomly coloring simple hypergraphs

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 Δ. (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<1α<1 we have Δ⩾nαΔ⩾nα and q⩾Δ(1+α)/kαq⩾Δ(1+α)/kα then Glauber dynamics will become close to uniform in O(nlogn) time from a random (improper) start. Note that for k>1+α−1k>1+α−1 we can take q=o(Δ)q=o(Δ).

► Analyses MCMC algorithm for randomly coloring a simple hypergraph. ► Works when number of available colors is o(Δ)o(Δ), where Δ is the maximum degree. ► Runs in O(nlogn) time.