Revisiting Hamiltonian Decomposition of the Hypercube

Abstract

A hypercube or binary n-cube is an interconnection network very suitable for implementing computing elements. In this paper we study the Hamiltonian decomposition, i.e. the partitioning of its edge set into Hamiltonian cycles. It is known that there are [n/2] disjoint Hamiltonian cycles on a binary n-cube. The proof of this result, however, does not give rise to any simple construction algorithm of such cycles. In a previous work Song [1995] presented ideas towards a simple method to this problem. First decompose the hypercube into cycles of length 16, C/sub 16/, and then apply a merge operator to join the C/sub 16/ cycles into larger Hamiltonian cycles. The case of dimension n=6 (a 64-node hypercube) is illustrated. He conjectures the method can be generalized for any even n. In this paper, we generalize the first phase of that method for any even n and prove its correctness. Also we show four possible merge operators for the case of n=8 (a 256-node hypercube). This result can be viewed as a step toward the general merge operator, thus proving the conjecture.