On the spanning connectivity and spanning laceability of hypercube-like networks

Let u and v be any two distinct nodes of an undirected graph G, which is k-connected. For 1 ≤ w ≤ k, a w-container C(u, v) of a k-connected graph G is a set of w-disjoint paths joining u and v. A w-container C(u, v) of G is a w-container if it contains all the nodes of G. A graph G is w-connected if there exists a w-container between any two distinct nodes. A bipartite graph G is w-laceable if there exists a w-container between any two nodes from different parts of G. In this study, we show that every graph in ndimensional bipartite graph B n is w -laceable for every w, 1 ≤ w ≤ n. It is shown that a constructed an n-dimensional non-bipartite graph H can not be 4-connected. In addition, we show that every graph in n-dimensional non-bipartite graph is w-connected for every w, 1 ≤ w ≤ 3.