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 [email protected][email protected]?k, a w-containerC(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. Let G"0=(V"0,E"0) and G"1=(V"1,E"1) be two disjoint graphs with |V"0|=|V"1|. Let E={(v,@f(v))|[email protected]?V"0,@f(v)@?V"1, and @f:V"0->V"1 is a bijection}. Let G=G"[email protected]?G"1=(V"[email protected]?V"1,E"[email protected]?E"[email protected]?E). The set of n-dimensional hypercube-like graph H"n^' is defined recursively as (a) H"1^'={K"2}, K"2= complete graph with two nodes, and (b) if G"0 and G"1 are in H"n^', then G=G"[email protected]?G"1 is in H"n"+"1^'. Let B"n^'={[email protected]?H"n^' and G is bipartite} and N"n^'=H"n^'@?B"n^'. In this paper, we show that every graph in B"n^' is w^*-laceable for every w, [email protected][email protected]?n. It is shown that a constructed N"n^'-graph H can not be 4^*-connected. In addition, we show that every graph in N"n^' is w^*-connected for every w, [email protected][email protected]?3.