Longest Fault-Free Paths in Hypercubes with both Faulty Nodes and Edges

Abstract

The hypercube is one of the most versatile and efficient interconnection networks for parallel computation. Let F_v (respectively, F_e) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Q_n. In this paper, we show that Q_n - F_v - F_e contains a fault free path with length at least 2ⁿ - 2|F_v| - 1(or 2ⁿ - 2|F_v| - 2) between two arbitrary vertices of odd (or even) distance if |F_v| + |F_e| les n - 2, where n ges 3. Since Q_n is bipartite of equal-size partite sets, the path is longest in the worst case. Furthermore, since Q_n is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free path obtained are worst-case optimal.