A two-stages tree-searching algorithm for finding three completely independent spanning trees

For the underlying graph G of a network, k (⩾2) spanning trees of G are called completely independent spanning trees (CISTs for short) if they are mutually inner-node-disjoint. It is well-known that determining the existence of k CISTs in a graph is an NP-hard problem, even for k=2. Also, there exists a k-connected graph which does not possess two CISTs for any k⩾2. Accordingly, researches focused on the problem of constructing multiple CISTs in some particular famous networks. Pai and Chang (2016) proposed a unified approach to recursively construct two CISTs with diameter 2n−1 in several n-dimensional hypercube-variant networks for n⩾4, including locally twisted cubes LTQn and crossed cubes CQn. Later on, new constructions showed that the diameter of two CISTs can be reduced to 2n−2 if n=4 and 2n−3 if n⩾5 for LTQn, and to 2n−2 if n=4,5 and 2n−3 if n⩾6 for CQn. In this paper, we intend to construct more CISTs for those networks that are paid attention. We develop a novel tree searching algorithm, called two-stages tree-searching algorithm (abbreviated as TS2), which can be used to construct three CISTs of a 6-regular graph if the scale of the graph is not too large and there exist three CISTs for certain. In particular, we demonstrate that three CISTs of LTQ6 and CQ6 can be acquired by using TS2. Moreover, for high-dimensional LTQn and CQn with n⩾7, we show that their three CISTs can be constructed by recursion. Consequently, we obtain the following results: (i) For LTQn, the diameters of three CISTs we constructed are 11, 12 and 12 when n=6, and all are 2n−1 when n⩾7; (ii) For CQn, all diameters of three CISTs we constructed are 12 when n=6, and 2n+1 when n⩾7.