  The bandwidth of a graph is the labeling of vertices with minimum maximum
edge difference. For many graph families this is NP-complete. A classic result
computes the bandwidth for the hypercube. We generalize this result to give
sharp lower bounds for products of cliques. This problem turns out to be
equivalent to one in communication over multiple channels in which channels can
fail and the information sent over those channels is lost. The goal is to
create an encoding that minimizes the difference between the received and the
original information while having as little redundancy as possible. Berger-Wolf
and Reingold [2] have considered the problem for the equal size cliques (or
equal capacity channels). This paper presents a tight lower bound and an
algorithm for constructing the labeling for the product of any number of
arbitrary size cliques.
