  An $L(h_1, h_2, \ldots, h_l)$-labelling of a graph $G$ is a mapping $\phi:
V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that for $1\le i\le l$ and each pair
of vertices $u, v$ of $G$ at distance $i$, we have $|\phi(u) - \phi(v)| \geq
h_i$. The span of $\phi$ is the difference between the largest and smallest
labels assigned to the vertices of $G$ by $\phi$, and $\lambda_{h_1, h_2,
\ldots, h_l}(G)$ is defined as the minimum span over all $L(h_1, h_2, \ldots,
h_l)$-labellings of $G$.
  In this paper we study $\lambda_{h, 1, \ldots, 1}$ for Cartesian products of
graphs, where $(h, 1, \ldots, 1)$ is an $l$-tuple with $l \ge 3$. We prove
that, under certain natural conditions, the value of this and three related
invariants on a graph $H$ which is the Cartesian product of $l$ graphs attain a
common lower bound. In particular, the chromatic number of the $l$-th power of
$H$ equals this lower bound plus one. We further obtain a sandwhich theorem
which extends the result to a family of subgraphs of $H$ which contain a
certain subgraph of $H$. All these results apply in particular to the class of
Hamming graphs: if $q_1\ge \cdots \ge q_d\ge 2$ and $3\le l\le d$ then the
Hamming graph $H=H_{q_1,q_2,\ldots ,q_d}$ satisfies
$\lambda_{q_l,1,\ldots,1}(H) = q_1q_2\ldots q_l-1$ whenever $q_1q_2\ldots
q_{l-1}>3(q_{l-1}+1)q_l\ldots q_d$. In particular, this settles a case of the
open problem on the chromatic number of powers of the hypercubes.
