Symmetric Properties and Two Variants of Shuffle-Cubes

Li et al. in [Inf. Process. Lett. 77 (2001) 35–41] proposed the shuffle-cube <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SQ_{n}$</tex-math></inline-formula>, 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula>-regular hypercube variants of dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> have diameter not less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n/2$</tex-math></inline-formula>. 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SQ_{n}$</tex-math></inline-formula> is not vertex-transitive for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n\gt 2$</tex-math></inline-formula>, 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SSQ_{n}$</tex-math></inline-formula> and balanced shuffle-cube <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$BSQ_{n}$</tex-math></inline-formula> are introduced, and their vertex-transitivity are proved simultaneously. By proposing the shuffle-cube-like graph, we obtain that both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SSQ_{n}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$BSQ_{n}$</tex-math></inline-formula> are maximally connected, implying high connectivity similar to the hypercube. Additionally, super-connectivity, a refined parameter of connectivity, of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SSQ_{n}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$BSQ_{n}$</tex-math></inline-formula> are also determined. Then, by vertex-transitivity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SSQ_{n}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$BSQ_{n}$</tex-math></inline-formula>, routing algorithms of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SSQ_{n}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$BSQ_{n}$</tex-math></inline-formula> are given for all <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n\gt 2$</tex-math></inline-formula> respectively. We show that both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SSQ_{n}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$BSQ_{n}$</tex-math></inline-formula> possess Hamiltonian cycle embedding for all <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n\gt 2$</tex-math></inline-formula>, and we also show that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SSQ_{n}$</tex-math></inline-formula> is Hamiltonian-connected. It is noticeable that each vertex of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$SSQ_{n}$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$BSQ_{n}$</tex-math></inline-formula>, we mend a flaw in the Property 3 in [IEEE Trans. Comput. 46 (1997) 484–490].