Hamiltonian path embedding and pancyclicity on the Mobius cube with faulty nodes and faulty edges

A graph G=(V, E) is said to be pancyclic if it contains fault-free cycles of all lengths from 4 to |V| in G. Let F_v and F_e be the sets of faulty nodes and faulty edges of an n-dimensional Mobius cube MQ_n, respectively, and let F=F_v∪F_e. A faulty graph is pancyclic if it contains fault-free cycles of all lengths from 4 to |V-F_v|. In this paper, we show that MQ_n-F contains a fault-free Hamiltonian path when |F|≤n-1 and n≥1. We also show that MQ_n-F is pancyclic when |F|≤n-2 and n≥2. Since MQ_n is regular of degree n, both results are optimal in the worst case.