Symmetric properties and two variants of shuffle-cubes

TL;DR

This paper investigates the symmetry of shuffle cubes, finds they are not vertex-transitive for n>2, and introduces two vertex-transitive variants with routing algorithms and Hamiltonian cycle properties.

Contribution

It introduces two new vertex-transitive variants of shuffle cubes, providing routing algorithms and Hamiltonian cycle embeddings, addressing symmetry limitations.

Findings

Shuffle cube $SQ_{n}$ is not vertex-transitive for all $n>2$.

The paper proposes simplified and balanced shuffle cubes $SSQ_{n}$ and $BSQ_{n}$.

Both variants have Hamiltonian cycle embeddings for all $n>2$.

Abstract

Li et al. in [Inf. Process. Lett. 77 (2001) 35--41] proposed the shuffle cube $SQ_{n}$ 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 $SQ_{n}$ 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 $SSQ_{n}$ and balanced shuffle cube $BSQ_{n}$ are introduced. Then, routing algorithms of $SSQ_{n}$ and $BSQ_{n}$ for all $n>2$ are given respectively. Furthermore, we show that both $SSQ_{n}$ and $BSQ_{n}$ 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].