Hybrid fault-tolerant prescribed hyper-hamiltonian laceability of hypercubes

The n-dimensional hypercube Q(n) is one of the most attractive interconnection networks for multiprocessor systems and it is a bipartite graph. Let F-v be a set of the end-nodes of kindependent edges in Q(n) and F-e be a set of fedges in Q(n)- F-v. Given a linear forest L of Q(n) - F-v - F-e, in this paper, we prove that (i) Q(n)-F-v-F-e admits a hamiltonian cycle passing through Lif vertical bar E(L)vertical bar + k + f = n - 2; and (ii) for any two nodes xand yof the opposite partite sets in Q(n) - F-v - F-e such that none of the paths in L has xor yas internal node or both of them as end-nodes, Q(n) - F-v - F-e admits a hamiltonian path between xand ypassing through Lif vertical bar E(L)vertical bar+ k + f <= n - 3; and (iii) for any two distinct nodes uand vof the partite set not containing win Q(n) - F-v - F-e - w such that none of the paths in Lhas uor vas internal node or both of them as end-nodes, Q(n) - F-v - F-e - wadmits a hamiltonian path between uand vpassing through Lif vertical bar E vertical bar(L)vertical bar + k + f <= n - 3, where wis an arbitrary node in Q(n)-F-v-F-e. (C) 2021 Elsevier B.V. All rights reserved.