Fault-free Ring Embedding in Faulty Crossed Cubes

Abstract

An n-dimensional crossed cube, CQn, is a variation from hypercube. In this paper, we prove that CQn is (n-2)-hamiltonian and (n-3)-hamiltonian connected. That is, a ring of length 2n - fv can be embedded in a faulty CQn with fv faulty nodes and fe faulty edges, where fv + fe ≦ n-2 and n ≧ 3. In other words, we show that the faulty CQn is still hamiltonian with n-2 faults. In addition, we also prove that there exists a hamiltonian path between any pair of vertices in a faulty CQn with n-3 faults. A recent result has shown that a ring of length 2n - 2fv can embedded in a faulty Hypercube, if fv + fe ≦ n-1 and n ≧ 4, with a few additional constrains [10]. Our results, in comparison to Hypercube, show that longer rings can be embedded in CQn without additional constrains.