Constructing Three Completely Independent Spanning Trees in Locally Twisted Cubes

Abstract

For the underlying graph G of a network, k spanning trees of G are called completely independent spanning trees (CISTs for short) if they are mutually inner-node-disjoint. It has been known that determining the existence of k CISTs in a graph is an NP-hard problem, even for $k=2$. Accordingly, researches focused on the problem of constructing multiple CISTs in some famous networks. Pai and Chang [28] proposed a unified approach to recursively construct two CISTs with diameter $2n-1$ in several n-dimensional hypercube-variant networks for $n\geqslant 4$, including locally twisted cubes $LTQ_n$. Later on, they provided a new construction for $LTQ_n$ and showed that the diameter of two CISTs can be reduced to $2n-2$ if $n=4$ (and thus is optimal) and $2n-3$ if $n\geqslant 5$. In this paper, we intend to construct more CISTs of $LTQ_n$. We develop a novel tree searching algorithm, called two-stages tree-searching algorithm, to construct three CISTs of $LTQ_6$ and show that the three CISTs of the high-dimensional $LTQ_n$ for $n\geqslant 7$ can be constructed by recursion. The diameters of three CISTs for $LTQ_n$ we constructed are 9, 12 and 14 when $n=6$, and are $2n-3$, $2n-1$ and $2n+1$ when $n\geqslant 7$.