On the connectivity of proper colorings of random graphs and hypergraphs

Abstract

Let Ωq=Ωq(H) denote the set of proper [q]‐colorings of the hypergraph H. Let Γq be the graph with vertex set Ωq where two colorings σ,τ are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=Hn,m;k, k ≥ 2, the random k‐uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Γq 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 Γq 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 Ωq is ergodic w.h.p.