On the Longest Fault-Free Paths in Hypercubes with More Faulty Nodes

Faults in a network may take various forms such as hardware/software errors, node/link faults, etc. In this paper, node-faults are addressed. Let F be a faulty set of f les 2n - 6 conditional node-faults in an injured n-cube Q n such that every node of Q n still has at least two fault - free neighbors. Then we show that Q n - F contains a path of length at least 2 n - 2f - 1 (respectively, 2 n - 2f - 2) between any two nodes of odd (respectively, even) distance. Since an n-cube is a bipartite graph, such kind of the fault- free path turns out to be the longest one in the case when all faulty nodes belong to the same partite set.