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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 2 n − 1 in several n -dimensional hypercube-variant networks for n ⩾ 4 , including locally twisted cubes L T Q n and crossed cubes C Q n. Later on, new constructions showed that the diameter of two CISTs can be reduced to 2 n − 2 if n = 4 and 2 n − 3 if n ⩾ 5 for L T Q n , and to 2 n − 2 if n = 4 , 5 and 2 n − 3 if n ⩾ 6 for C Q 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 TS 2), 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 L T Q 6 and C Q 6 can be acquired by using TS 2. Moreover, for high-dimensional L T Q n and C Q n with n ⩾ 7 , we show that their three CISTs can be constructed by recursion. Consequently, we obtain the following results: (i) For L T Q n , the diameters of three CISTs we constructed are 11, 12 and 12 when n = 6 , and all are 2 n − 1 when n ⩾ 7 ; (ii) For C Q n , all diameters of three CISTs we constructed are 12 when n = 6 , and 2 n + 1 when n ⩾ 7. [ABSTRACT FROM AUTHOR]

Given a graph G , a set of spanning trees of G are completely independent spanning trees (CISTs for short) if for any two vertices x , y ∈ V ( G ) , the paths joining x and y on these trees have neither vertex nor edge in common, except x and y . Hasunuma [9,10] first introduced the concept of CISTs and conjectured that there are k CISTs in any 2 k -connected graph. Unfortunately, Péterfalvi [16] disproved this conjecture by constructing a k -connected graph, for each k ⩾ 2 , which does not have two CISTs. In this paper, we provide a unified approach for constructing two CISTs in several hypercube-variant networks, including hypercubes, locally twisted cubes, crossed cubes, parity cubes, and Möbius cubes. In particular, for an n -dimensional hypercube-variant network, the diameters of the constructed CISTs are 2 n − 1 . [ABSTRACT FROM AUTHOR]

• We investigate the problem of constructing a dual-CIST on bicubes B Q n. • A newly generic adjustment scheme can reduce the diameter of dual-CIST. • The diameter of the constructed dual-CIST of B Q n is 2 n − 2 for n ⩾ 5 odd. • The diameter of the constructed dual-CIST of B Q n is 2 n − 3 for n ⩾ 6 even. • As a by-product, a folded hypercube F Q n admits a dual-CIST with diameter 2 n − 2. Recently, an innovative hypercube-variant network called bicube, denoted as B Q n , has been proposed to possess both short diameter and symmetry advantages. Unlike other existing hypercube-variant networks, they lose their symmetry in pursuit of short diameters. For solving the problems of fault-tolerant transmission and secure message distribution in a reliable network, one solution suggested a dual-CIST (two completely independent spanning trees) to design a multi-path routing (e.g., a recently proposed secure-protection routing). We can make the construction using the standard arrangement guideline (SAG) like the hypercubes to obtain a dual-CIST with a diameter of 2 n − 1 on B Q n. This paper proposes a newly generic adjustment scheme (GAS) for reducing the diameter of the dual-CIST under this construction. As a result, the diameter of T i for i = 1 , 2 we constructed for B Q n are as follows: diam (T i) = { 7 if n = 4 ; 2 n − 2 if n ⩾ 5 and n is odd ; 2 n − 3 if n ⩾ 6 and n is even. [ABSTRACT FROM AUTHOR]