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 hypercubevariant networks for n >= 4, including locally twisted cubes LTQ(n) and crossed cubes CQ(n). 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 LTQ(n) and to 2n - 2 if n = 4, 5 and 2n - 3 if n >= 6 for CQ(n). 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 LT Q(6) and CQ(6) can be acquired by using TS2. Moreover, for high-dimensional L T Q(n) and CQ(n) with n >= 7, we show that their three CISTs can be constructed by recursion. Consequently, we obtain the following results: (i) For LT Q(n), the diameters of three CISTs we constructed are 11, 12 and 12 when n = 6, and all are 2n - I when n >= 7; (ii) For CQ(n), all diameters of three CISTs we constructed are 12 when n = 6, and 2n +1 when n >= 7. (C) 2019 Elsevier B.V. All rights reserved.