Fault-tolerant hamiltonian laceability of hypercubes

Abstract

It is known that every hypercube Q(n) is a bipartite graph. Assume that n greater than or equal to 2 and F is a subset of edges with F less than or equal to n - 2. We prove that there exists a hamiltonian path in Q(n) - F between any two vertices of different partite sets. Moreover, there exists a path of length 2(n) - 2 between any two vertices of the same partite set. Assume that n greater than or equal to 3 and F is a subset of edges with F less than or equal to n - 3. We prove that there exists a hamiltonian path in Q(n) - {v} - F between any two vertices in the partite set without v. Furthermore, all bounds are tight. (C) 2002 Elsevier Science B.V. All rights reserved.