Symmetric properties and two variants of shuffle-cubes

Li et al. in [Inf. Process. Lett. 77 (2001) 35--41] proposed the shuffle cube SQₙ as an attractive interconnection network topology for massive parallel and distributed systems. By far, symmetric properties of the shuffle cube remains unknown. In this paper, we show that SQₙ is not vertex-transitive for all $n>2$, which is not an appealing property in interconnection networks. To overcome this limitation, two novel vertex-transitive variants of the shuffle-cube, namely simplified shuffle-cube SSQₙ and balanced shuffle cube BSQₙ are introduced. Then, routing algorithms of SSQₙ and BSQₙ for all $n>2$ are given respectively. Furthermore, we show that both SSQₙ and BSQₙ possess Hamiltonian cycle embedding for all $n>2$. Finally, as a by-product, we mend a flaw in the Property 3 in [IEEE Trans. Comput. 46 (1997) 484--490].