On the connectivity threshold for colorings of random graphs and hypergraphs

Let $\Omegaq=\Omegaq(H)$ denote the set of proper $[q]$-colorings of the hypergraph $H$. Let $\Gammaq$ be the graph with vertex set $\Omegaq$ and an edge ${\sigma,\tau}$ where $\sigma,\tau$ are colorings iff $h(\sigma,\tau)=1$. Here $h(\sigma,\tau)$ is the Hamming distance $|{v\in V(H):\sigma(v)\neq\tau(v)}|$. We show that if $H=H{n,m;k},\,k\geq 2$, the random $k$-uniform hypergraph with $V=[n]$ and $m=dn/k$ then w.h.p. $\Gammaq$ is connected if $d$ is sufficiently large and $q\gtrsim (d/\log d)^{1/(k-1)}$.