Mutually independent Hamiltonian cycles in dual-cubes

Abstract

The hypercube family Q _n is one of the most well-known interconnection networks in parallel computers. With Q _n , dual-cube networks, denoted by DC _n , was introduced and shown to be a (n+1)-regular, vertex symmetric graph with some fault-tolerant Hamiltonian properties. In addition, DC _n ’s are shown to be superior to Q _n ’s in many aspects. In this article, we will prove that the n-dimensional dual-cube DC _n contains n+1 mutually independent Hamiltonian cycles for n≥2. More specifically, let v _i ∈V(DC _n ) for 0≤i≤|V(DC _n )|−1 and let $\langle v_{0},v_{1},\ldots ,v_{|V(\mathit{DC}_{n})|-1},v_{0}\rangle$ be a Hamiltonian cycle of DC _n . We prove that DC _n contains n+1 Hamiltonian cycles of the form $\langle v_{0},v_{1}^{k},\ldots,v_{|V(\mathit{DC}_{n})|-1}^{k},v_{0}\rangle$ for 0≤k≤n, in which v ^k _i ≠v ^k′ _i whenever k≠k′. The result is optimal since each vertex of DC _n has only n+1 neighbors.