Hamiltonian Connectedness of Recursive Dual-Net

Recursive Dual-Net (RDN) was proposed recently as an effective, high-performance interconnection network for supercomputers with millions of nodes. A Recursive Dual-Net RDN (B) is recursively constructed on a base symmetric network B. At each iteration, the network is extended through dual-construction. The dual-construction extends a symmetric graph G into a symmetric graph G´ with size 2n² and node-degree d + 1, where n and d are the size and the node-degree of G, respectively. Therefore, a k-level Recursive Dual-Net RDN^k (B) contains (2n₀)^2k/2 nodes with a node degree of d₀ + k, where n₀ and d₀ are the size and the node-degree of the base network B, respectively. In this paper, we show that, if the base network B is hamiltonian, RDN^k (B) is hamiltonian. We give an efficient algorithm for constructing a hamiltonian cycle in RDN^k (B) for k > 0. We also show that if the base network is hamiltonian connected, RDN^k (B) is hamiltonian connected for any k > 0.