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 $\bm{RDN(B)}$ is recursively constructed on a base symmetric network $\bm{B}$. At each iteration, the network is extended through dual-construction. The dual-construction extends a symmetric graph $\bm{G}$ into a symmetric graph $\bm{G'}$ with size $\bm{2n^2}$ and node-degree $\bm{d+1}$, where $\bm{n}$ and $\bm{d}$ are the size and the node-degree of $\bm{G}$, respectively. Therefore, a $\bm{k}$-level Recursive Dual-Net $\bm{RDN^k(B)}$ contains $\bm{(2n_0)^{2^k}/2}$ nodes with a node degree of $\bm{d_0+k}$, where $\bm{n_0}$ and $\bm{d_0}$ are the size and the node-degree of the base network $\bm{B}$, respectively. In this paper, we show that, if the base network $\bm{B}$ is hamiltonian, $\bm{RDN^k(B)}$ is hamiltonian. We give an efficient algorithm for constructing a hamiltonian cycle in $\bm{RDN^k(B)}$ for $\bm{k≫0}$. We also show that if the base network is hamiltonian connected, $\bm{RDN^k(B)}$ is hamiltonian connected for any $\bm{k≫0}$.