On the fault-tolerant pancyclicity of crossed cubes

A graph is pancyclic if it contains all cycles from lengths 4 to |V(G)|. An n-dimensional crossed cube, an important variation of hypercube denoted as CQ_n, has been proved to be pancyclic because it contains all cycles whose lengths range from 4 to |V(CQ_n)|. Since vertex and edge faults may occur when a network is used, it is practical and meaningful to evaluate the performance of a faulty network. Moreover the vertex fault-tolerant Hamiltonicity and the edge fault-tolerant Hamiltonicity measure the performances of the Hamiltonian properties in the faulty networks. From this fault-tolerant concept, we propose using the fault-tolerant pancyclicity of networks to measure the performance of faulty networks. In this paper we consider a faulty crossed n-cube with vertex and/or edge faults here. Let the faulty set F be a subset of V(CQ_n)∪E(CQ_n). We prove that any cycle of length l(4≤l≤|V(CQ_n)|-f_ν) can be embedded into a faulty crossed n-cube CQ_n-F with dilation 1, where |F|=f_ν+f_e is less than n-2, f_ν is the number of faulty vertices of F, f_e is the number of faulty edges of F, and n is greater than 2. The results can readily be used in the optimum embedding of a ring of the specified length in a faulty crossed cube.