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 i k ?v i k? whenever k?k?. The result is optimal since each vertex of DC n has only n+1 neighbors.