  We establish an inclusion relation between two uniform models of random
$k$-graphs (for constant $k \ge 2$) on $n$ labeled vertices: $\mathbb
G^{(k)}(n,m)$, the random $k$-graph with $m$ edges, and $\mathbb R^{(k)}(n,d)$,
the random $d$-regular $k$-graph. We show that if $n\log n\ll m\ll n^k$ we can
choose $d = d(n) \sim {km}/n$ and couple $\mathbb G^{(k)}(n,m)$ and $\mathbb
R^{(k)}(n,d)$ so that the latter contains the former with probability tending
to one as $n\to\infty$. This extends an earlier result of Kim and Vu about
"sandwiching random graphs". In view of known threshold theorems on the
existence of different types of Hamilton cycles in $\mathbb G^{(k)}(n,m)$, our
result allows us to find conditions under which $\mathbb R^{(k)}(n,d)$ is
Hamiltonian. In particular, for $k\ge 3$ we conclude that if $n^{k-2} \ll d \ll
n^{k-1}$, then a.a.s. $\mathbb R^{(k)}(n,d)$ contains a tight Hamilton cycle.
