On container length and connectivity in unidirectional hypercubes.

In this paper, we show that the two unidirectional binary n-cubes, namely, Q(1)(n) and Q(2)(n), proposed as high-speed networking schemes by Chou and Du are maximum fault-tolerant, that is, from vertex a to vertex b in the network, we can determine a set of zeta(a, b) vertex-disjoint routing paths, where zeta(a, b) = inverted right perpendicular n/2 inverted left perpendicular, if a has even parity and b has odd parity and right perpendicular n/2 left perpendicular otherwise. Furthermore, the container problem as defined by Hsu for the two maximum fault-tolerant, unidirectional topologies is studied. In particular, we show that the smallest possible length for any maximum fault-tolerant container from a to b is at most (1) l + 4, where l is the shortest path length in Q(1)(n) from a to b, and (2) l + 5, where l is the shortest path length in Q(2)(n) (n is odd) (i) from a to b when a and b have the same leading-bit values and (ii) from a to b' (b' and b only differ at leading-bit positions) when otherwise. (C) 1998 John Wiley & Sons, Inc.