On ring embedding in hypercubes with faulty nodes and links

Abstract

In this paper, we show that in an n-dimensional hypercube Q n with f n faulty nodes and f e faulty edges, such that f n + f e ⩽ n − 1, a ring of length 2 n − 2 f n can be embedded avoiding the faulty elements when f n > 0 or f n < n − 1. When f n = 0 and f e = n − 1, if all the faulty edges are not incident on the same node, a Hamiltonian cycle can be embedded avoiding the faulty elements when n ⩾ 4. For a Q 3, however, if f n = 0 and f e = 2, a Hamiltonian cycle might not exist even when all faulty edges are not incident on the same node. We show that a ring of size 6 can be embedded in that case. When f n = 0 and f e = n − 1, if all the faulty edges are incident on the same node, clearly a Hamiltonian cycle cannot exist and we show that a ring of size 2 n − 2 can be embedded. This generalizes a recent result of Tseng (1996) where the number of edge faults were assumed not to exceed n − 4.