Symmetric Properties and Two Variants of Shuffle-Cubes

Li et al. in [Inf. Process. Lett. 77 (2001) 35–41] proposed the shuffle-cube $SQ_{n}$, a hypercube variant, as an attractive interconnection network topology for massive parallel and distributed systems. Diameter and symmetry are two desirable measures of network performance in terms of transmission delay and routing algorithms. Almost all $n$-regular hypercube variants of dimension $n$ have diameter not less than $n/2$. The diameter of the shuffle-cube is approximately a quarter of the diameter of the hypercube of the same dimension, making it a competitive candidate network topology. By far, symmetric properties of the shuffle-cube remain unknown. In this paper, we show that $SQ_{n}$ is not vertex-transitive for $n> 2$, which is not an appealing property in interconnection networks. This shortcoming limits the practical application of the shuffle-cube. To overcome this limitation, two novel variants of the shuffle-cube, namely simplified shuffle-cube $SSQ_{n}$ and balanced shuffle-cube $BSQ_{n}$ are introduced, and their vertex-transitivity are proved simultaneously. By proposing the shuffle-cube-like graph, we obtain that both $SSQ_{n}$ and $BSQ_{n}$ are maximally connected, implying high connectivity similar to the hypercube. Additionally, super-connectivity, a refined parameter of connectivity, of $SSQ_{n}$ and $BSQ_{n}$ are also determined. Then, by vertex-transitivity of $SSQ_{n}$ and $BSQ_{n}$, routing algorithms of $SSQ_{n}$ and $BSQ_{n}$ are given for all $n> 2$ respectively. We show that both $SSQ_{n}$ and $BSQ_{n}$ possess Hamiltonian cycle embedding for all $n> 2$, and we also show that $SSQ_{n}$ is Hamiltonian-connected. It is noticeable that each vertex of $SSQ_{n}$ is contained in exactly one clique of size four, making it also a viable interconnection topology for data center networking since each clique of size four can be viewed as an efficient local data processing cluster of the network. Finally, as a by-product of proving vertex-transitivity of $BSQ_{n}$, we mend a flaw in the Property 3 in [IEEE Trans. Comput. 46 (1997) 484–490].