The spanning connectivity of folded hypercubes

Abstract

A k-container of a graph G is a set of k internally disjoint paths between u and v. A k-container of G is a k∗-container if it contains all vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices, and a bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u and v from different partite sets of G for a given k. A k-connected graph (respectively, bipartite graph) G is f-edge fault-tolerant spanning connected (respectively, laceable) if G − F is w∗-connected for any w with 1 ⩽ w ⩽ k − f and for any set F of f faulty edges in G. This paper shows that the folded hypercube FQ n is f-edge fault-tolerant spanning laceable if n(⩾3) is odd and f ⩽ n − 1, and f-edge fault-tolerant spanning connected if n (⩾2) is even and f ⩽ n − 2.