Hamiltonian connectivity and globally 3∗-connectivity of dual-cube extensive networks

In 2000, Li et al. introduced dual-cube networks, denoted by DCnDCn for n⩾1n⩾1, using the hypercube family QnQn and showed the vertex symmetry and some fault-tolerant hamiltonian properties of DCnDCn. In this article, we introduce a new family of interconnection networks called dual-cube extensive networks, denoted by DCEN(G)DCEN(G). Given any arbitrary graph GG, DCEN(G)DCEN(G) is generated from GG using the similar structure of DCnDCn. We show that if GG is a nonbipartite and hamiltonian connected graph, then DCEN(G)DCEN(G) is hamiltonian connected. In addition, if GG has the property that for any two distinct vertices u,vu,v of GG, there exist three disjoint paths between uu and vv such that these three paths span the graph GG, then DCEN(G)DCEN(G) preserves the same property. Furthermore, we prove that the similar results hold when GG is a bipartite graph.