Completely Independent Spanning Trees on Some Interconnection Networks

Abstract

SUMMARYLet T 1 , T 2 , . . . , T k be spanning trees in a graph G. If, for any two vertices u, v of G, the paths joining u and v on the k trees are mutually vertex-disjoint, then T 1 , T 2 , . . . , T k are called completely independent spanning trees (CISTs for short) of G. The construction of CISTs can be applied in fault-tolerant broadcasting and secure message distribution on interconnection networks. Hasunuma (2001) first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Unfortunately, this conjecture was disproved by Péterfalvi recently. In this note, we give a necessary condition for k-connected k-regular graphs with k/2 CISTs. Based on this condition, we provide more counterexamples for Hasunuma's conjecture. By contrast, we show that there are two CISTs in 4-regular chordal rings CR (N, d) with N = k(d − 1) + j under the condition that k 4 is even and 0 j 4. In particular, the diameter of each constructed CIST is derived.

Constructing Two Completely Independent Spanning Trees in Balanced Hypercubes

“…For constructing two CISTs, the results in [5], [13], [14], [23] are based on Theorem 1, and the results in [2], [21] are based on Theorem 2. In this paper, we apply Theorem 1 to construct two CISTs of balanced hypercubes.…”

“…Accordingly, researches investigating sufficient conditions for graphs that admit multiple CISTs, such as degree-based conditions, can be found in [1], [3], [7], [15], [17]. Also, with the help of constructions, it has been confirmed that certain classes of graphs possess two CISTs, e.g., 4-connected maximal planar graphs [13], Cartesian product of any 2-connected graphs [14], 4-regular chordal rings [2], [23], crossed cubes [5], and several hypercube-variant networks [21]. In addition, more graphs possessing multiple CISTs can be found in [6], [12], [16], [19], [20], [22].…”

Construction of Four Completely Independent Spanning Trees on Augmented Cubes

“…, 00001(2), 00010(3), 00011(4), 00100(5), 00101 (6), 00110 (7), 00111 (8), 01000 (9), 01001 (10), 01010 (11), 01011 (12), 01100 (13), 01101 (14), 01110 (15), 01111 (16), 10000 (17), 10001 (18), 10010 (19) 18) 11000( 25) 10000( 17) 00000 (1) 01111( 16) 01100( 13) 01101( 14) 01110( 15) 00010( 3) 00001( 2) 00100( 5) 10100( 21) 11011( 28) 00011( 4) 01011( 12) 00110( 7) 00101( 6) 00111( 8) 01001( 10 18) 00110( 7) 10100( 21) 10000 (17) 10011( 20) 11011( 28) 01001 (10) 01110( 15) 01000( 9) 01010( 11) 00001( 2) 00101( 6) 00010( 3) 00011( 4) 01011( 12) 01100( 13) 00000( 1) 11100( 29) 00111( 8) 00100…”

“…He showed that there exists a k−connected graph which does not contain two CISTs for each k ≥ 2. Pai et al [14] showed that the results are negative to Hasunuma's conjecture in case of hypercube of dimension n ∈ {10, 12, 14, 20, 22, 24, 26, 28, 30}. Many authors provided a necessary condition of CISTs [1,2,5,9,12].…”

The Construction of Multiple Independent Spanning Trees on Burnt Pancake Networks

“…He also posted a conjecture that there exist k CISTs in a 2k-connected graph. Currently, this conjecture has been proved to fail by counterexamples [23], [28]. For recent research results on CISTs and their applications, the reader can refer to [24]- [27] and references quoted therein.…”