Hamiltonicity of the Hierarchical Cubic Network

Abstract.

The hierarchical cubic network was proposed as an alternative to the hypercube. In this paper, using Gray codes, we show that the hierarchical cubic network is hamiltonian-connected. A network is hamiltonian-connected if it contains a hamiltonian path between every two distinct nodes. In other words, a hamiltonian-connected network can embed a longest linear array between every two distinct nodes with dilation, congestion, load, and expansion all equal to 1. We also show that the hierarchical cubic network contains cycles of all possible lengths except 3 and 5. Since the hypercube contains cycles only of even lengths, it is concluded that the hierarchical cubic network is superior to the hypercube in hamilton- icity. Our results can be applied to the hierarchical folded-hypercube network as well.