Hybrid fault-tolerant prescribed hyper-hamiltonian laceability of hypercubes

Abstract

The n-dimensional hypercube Qn is one of the most attractive interconnection networks for multiprocessor systems and it is a bipartite graph. Let Fv be a set of the end-nodes of k independent edges in Qn and Fe be a set of f edges in Qn−Fv. Given a linear forest L of Qn−Fv−Fe, in this paper, we prove that (i) Qn−Fv−Fe admits a hamiltonian cycle passing through L if |E(L)|+k+f≤n−2; and (ii) for any two nodes x and y of the opposite partite sets in Qn−Fv−Fe such that none of the paths in L has x or y as internal node or both of them as end-nodes, Qn−Fv−Fe admits a hamiltonian path between x and y passing through L if |E(L)|+k+f≤n−3; and (iii) for any two distinct nodes u and v of the partite set not containing w in Qn−Fv−Fe−w such that none of the paths in L has u or v as internal node or both of them as end-nodes, Qn−Fv−Fe−w admits a hamiltonian path between u and v passing through L if |E(L)|+k+f≤n−3, where w is an arbitrary node in Qn−Fv−Fe.