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

Abstract

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/sub v/ and F/sub e/ be the sets of faulty nodes and faulty edges of an n-dimensional Mobius cube MQ/sub n/, respectively, and let F=F/sub v//spl cup/F/sub e/. A faulty graph is pancyclic if it contains fault-free cycles of all lengths from 4 to |V-F/sub v/|. In this paper, we show that MQ/sub n/-F contains a fault-free Hamiltonian path when |F|/spl les/n-1 and n/spl ges/1. We also show that MQ/sub n/-F is pancyclic when |F|/spl les/n-2 and n/spl ges/2. Since MQ/sub n/ is regular of degree n, both results are optimal in the worst case.