Fault-Tolerant Mutually Independent Hamiltonian Cycles Embedding on Hypercubes

A Hamiltonian path (respectively, cycle) in G is a path (respectively, cycle) which contains every vertex of G exactly once. Two Hamiltonian paths in a graph G, P₁=langu₁, u₂,..., u_nrang and P₂=langv₁ , v₂,..., v_nrang, are full-independent if u_inev_i for every 1lesilesn. A set of Hamiltonian paths {P₁, P₂,..., P_k} of G are mutually full-independent if any two different Hamiltonian paths in the set are full-independent. On the other hand, two Hamiltonian cycles, C₁=langu₁, u₂,..., u_n, u ₁rang and C₂=langv₁, v₂,..., v_n, v₁rang, are independent starting at u₁ if u₁=v₁ and u_i nev_i for every 1<ilesn. A set of Hamiltonian cycles {C₁, C₂,..., C_k} of G are mutually independent starting at v if any two different Hamiltonian cycles in the set are independent starting at v. Let F be the set of faulty edges of Q_n such that 1les|F|lesn-2. In this paper, we show that Q_n-F contains n-|F|-1 fault-free mutually full-independent Hamiltonian paths between two adjacent vertices, where nges3. We also show that Q_n contains n-|F|-1 fault-free mutually independent Hamiltonian cycles starting at any vertex, where nges4