On the connectivity of proper colorings of random graphs and hypergraphs.

Let Omega(q) = Omega(q)(H) denote the set of proper [q]-colorings of the hypergraph H. Het Gamma(q) be the graph with vertex set Omega(q) where two colorings sigma, tau are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H = H-n,H-m;k, k >= 2, the random k-uniform hypergraph with V = [n] and m = dn/k hyperedges then w.h.p. Gamma(q) is connected if d is sufficiently large and q greater than or similar to (d/log d)(1/(k-1)). This is optimal up to the first order in d. Furthermore, with a few more colors, we find that the diameter of Gamma(q) is O(n) w.h.p., where the hidden constant depends on d. So, with this choice of d, q, the natural Glauber dynamics Markov Chain on Omega(q) is ergodic w.h.p.