The Fault-Tolerant Hamiltonian Problems of Crossed Cubes with Path Faults

In this paper, we investigate the fault-tolerant Hamiltonian problems of crossed cubes with a faulty path. More precisely, let P denote any path in an n-dimensional crossed cube CQ(n) for n >= 5, and let V(P) be the vertex set of P. We show that CQ(n) - V(P) is Hamiltonian if vertical bar V(P)vertical bar <= n and is Hamiltonian connected if vertical bar V(P)vertical bar <= n-1. Compared with the previous results showing that the crossed cube is (n - 2)-fault-tolerant Hamiltonian and (n - 3)-fault-tolerant Hamiltonian connected for arbitrary faults, the contribution of this paper indicates that the crossed cube can tolerate more faulty vertices if these vertices happen to form some specific types of structures.