Constructing tri-CISTs in shuffle-cubes

Abstract

Let ${\mathcal {T}}=\{T_1,T_2,\ldots ,T_k\}$ be a set of $k\geqslant 2$ spanning trees in a graph G. The k spanning trees are called completely independent spanning trees (CISTs for short) if the paths joining every pair of vertices x and y in any two trees have neither vertex nor edge in common except for x and y. Particularly, ${\mathcal {T}}$ is called a dual-CIST (resp. tri-CIST) provided $k=2$ (resp. $k=3$). From an algorithmic point of view, the problem of finding a dual-CIST in a given graph is known to be NP-hard. For data transmission applications in reliable networks, the existence of a dual-CIST can provide a configuration of fault-tolerant routing called protection routing. The presence of a tri-CIST can enhance the dependability of transmission and carry out secure message distribution for confidentiality. Recently, the construction of a dual-CIST has been proposed in a shuffle-cube $SQ_n$, which is an innovative hypercube-variant network that possesses both short diameter and connectivity advantages. This paper uses the CIST-partition technique to construct a tri-CIST of $SQ_6$, and shows that the diameters of three CISTs are 22, 22, and 13. Then, by the hierarchical structure of $SQ_n$, we propose a recursive algorithm for constructing a tri-CIST for high-dimensional shuffle-cubes. When $n\geqslant 10$, the diameters of $T_i$, $i=1,2,3$, we constructed for $SQ_n$ are as follows: $2n+11$, $2n+9$, and $2n+1$.