The dual-cube is a newly proposed topology for interconnection networks, which uses low dimensional hypercubes as building blocks. The primary advantages of the dual-cube over the hypercube are that, with the same node degree n, the dual-cube contains 2^n-1 times more nodes than the hypercube and, with the same amount of nodes, the dual-cube has approximately 50% less links than the hypercube. This paper was focused on the investigations of the structural self-similarity and the Hamiltonian property of the dual-cube. It was shown that a dual-cube can be recursively constructed from lower dimensional dual-cubes and, conversely, a dual-cube can be recursively decomposed into lower dimensional dual-cubes. It was also proved that all dual-cubes are Hamiltonian. There exist multiple Hamiltonian cycles on a dual-cube, among which, \left\lfloor {{\raise0.7ex\hbox{${(n - 1)}$} \!\mathord{\left/ {\vphantom {{(n - 1)} 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} \right\rfloor cycles are edge-disjoint. It was illustrated that rings and linear arrays can be effectively emulated on dual-cubes. Some strategies for track sharing in efficient VLSI layout design were also discussed.