On the connectivity of proper colorings of random graphs and hypergraphs

Let Ω_q=Ω_q(H) denote the set of proper [q]‐colorings of the hypergraph H. Let Γ_q be the graph with vertex set Ω_q where two colorings σ,τ are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_n,m;k, k ≥ 2, the random k‐uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Γ_q is connected if d is sufficiently large and . This is optimal up to the first order in d. Furthermore, with a few more colors, we find that the diameter of Γ_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 Ω_q is ergodic w.h.p.