The super laceability of the hypercubes

A k-container C(u,v) of a graph G is a set of k disjoint paths joining u to v. A k-container C(u,v) is a k∗-container if every vertex of G is incident with a path in C(u,v). A bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u, v from different partite set of G. A bipartite graph G with connectivity k is super laceable if it is i∗-laceable for all i k. A bipartite graph G with connectivity k is f -edge fault-tolerant super laceable if G− F is i∗-laceable for any 1 i k − f and for any edge subset F with |F | = f < k − 1. In this paper, we prove that the hypercube graph Qr is super laceable. Moreover, Qr is f -edge fault-tolerant super laceable for any f r − 2.  2004 Elsevier B.V. All rights reserved.