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

Abstract

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. Let G(0) = (V-0, E-0) and G(1) = (V-1, E-1) be two disjoint graphs with vertical bar V-0 vertical bar= vertical bar V-1 vertical bar. Let E = ((v, phi(v)) vertical bar v is an element of V0, phi(v) is an element of V-1, and phi : V-0 -> V-1 is a bijection}. Let G = G(0) circle plus G(1) = (V0 boolean OR V1, E-0 boolean OR E-1 boolean OR E). The set of n-dimensional hypercube-like graph H-n(') is defined recursively as (a) H-1(') = {K2}, K2 = complete graph with two nodes, and (b) if G(0) and G(1) are in H-n', then G = G(0) circle plus G(1) is in H-n+1('). Let B-n' = {G is an element of 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, 1 <= w <= 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, 1 <= w <= 3. (c) 2007 Published by Elsevier B.V.