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.

We shall define the k-isoperimetric connectivity $\lambda _k$ of a regular graph $\Gamma $ as the minimum number of arcs originating in a set with cardinality not exceeding half the order of the graph and containing at least k vertices. Clearly $\lambda _k \leq dk-e_k$, where d is the degree of $\Gamma$ and $e_k$ is the maximal number of edges induced on a set of k vertices. We shall show that Cayley graphs with a prime order and arc-transitive graphs have $\lambda _k =dk-e_k$, provided that $d\geq 3k-3$. We describe all vertex-transitive graphs where $\lambda _2\leq 2d-3$.

The following problem arises in the design of some interconnection networks for distributed systems. Namely, to construct digraphs with given maximum out-degree, reduced diameter, easy routing, good connectivity, and good expandability. To this end, a method based on the concept of partial line digraph is presented. This proposal, which turns out to be a generalization of the so-called line digraph technique, allows digraphs that satisfy all the above-mentioned requirements to be obtained. In particular, it is shown that the partial line digraphs of Kautz digraphs solve the (d, N) digraph problem, i.e. to minimize the diameter D in a digraph of maximum out-degree d and number of vertices N, for any N in the range d/sup D-1/+d/sup D-2/+. . .+1>