Completely independent spanning trees in some regular graphs

Abstract

International audienceLet k >= 2 be an integer and T-1,..., T-k be spanning trees of a graph G. If for any pair of vertices {u, v} of V(G), the paths between u and v in every T-i, 1 <= i <= k, do not contain common edges and common vertices, except the vertices u and v, then T1,... Tk are completely independent spanning trees in G. For 2k-regular graphs which are 2k-connected, such as the Cartesian product of a complete graph of order 2k-1 and a cycle, and some Cartesian products of three cycles (for k = 3), the maximum number of completely independent spanning trees contained in these graphs is determined and it turns out that this maximum is not always k. (C) 2016 Elsevier B.V. All rights reserved

Completely independent spanning trees for enhancing the robustness in ad-hoc Networks

“…The search for CISTs gained inceasing interest in 2012 when Péterfalvi disproved the conjecture [23] by proposing a construction of a k-connected graph that does not contain 2 CISTs, for any k ≥ 0, thus showing the problem was much more difficult than it seemed to be. A similar work showed that even for 2k-connected and 2k-regular graphs, the maximum number of CISTs is not always k [24].…”

Constructing Two Completely Independent Spanning Trees in Balanced Hypercubes

“…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]. In this paper, we investigate the problem of constructing two CISTs in balanced hypercubes (defined later in Sect.…”

Dual Protection Routing Trees on Graphs

“…Vertices 0, 2, 3, and 4 are inner nodes in T1, and vertices 1, 5, 6, and 7 are inner nodes in T2. Stems are (0, 2), (0, 4), (2, 3) (respectively, (1, 5), (5, 7), (6, 7)), and leaf edges are (1, 3), (3,7), (2,6), (4, 5) (respectively, (0, 1), (2,6), (3,7), (4,5)) in T1 (respectively, T2). Clearly, T1 and T2 obey Definitions 4 and 6.…”

“…Vertices 1, 2, 6, 7, 10, 11, 13, and 14 are inner nodes in T 1 , and vertices 0, 3, 4, 5, 8, 9, 12, 15 are inner nodes in T 2 . Stems are (1, 7), (1,13), (2,6), (6, 7), (6,14), (7,11), (10,11) (respectively, (0, 4), (0, 8), (3,5), (4,5), (4,12), (8,9), (9,15)) and leaf edges are (0, 2), (1, 3), (4,6), (5,7), (8,10), (9,11), (12,14), (13,15) (respectively, (0, 1), (2, 3), (4,6), (5,7), (8,10), (9,11), (12,14), (13,15)) in T 1 (respectively, T 2 ). Clearly, T 1 and T 2 obey Definitions 4 and 6.…”

“…Cheng et al constructed two CISTs in crossed cubes [6]. Darties et al determined three CISTs in some Cartesian products of three cycles [7]. Pai et al proposed several known CISTs, such as complete graphs, complete bipartite, complete tripartite graphs [8], and variants of hypercubes [9].…”