Constructing two completely independent spanning trees in hypercube-variant networks

Constructing Two Completely Independent Spanning Trees in 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].…”

“…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.…”

“…Then G n can be composed from G 0 n−1 and G 1 n−1 by adding appropriate edges in 2 n−1 disjoint pairs of vertices between G 0 n−1 and G 1 n−1 (i.e., a perfect matching). Hence, Pai and Chang [21] proposed a unified approach to recursively construct two CISTs in several hypercube-variant networks, including hypercubes, locally twisted cubes, crossed cubes, parity cubes, and Möbius cubes. This unified approach is based on the following theorem.…”

“…This unified approach is based on the following theorem. Theorem 3: [21] For n 5 and i ∈ {0, 1}, let T i 1 and T i 2 be two CISTs of G i n−1 . Then, two CISTs of G n , sayT j for j ∈ {1, 2}, can be constructed from T 0 j and T 1 j by adding an edge (u j , v j ) ∈ E(G n ) to connect two inner-vertices u j ∈ V(T 0 j ) and v j ∈ V(T 1 j ).…”

Construction of Four Completely Independent Spanning Trees on Augmented Cubes

“…Constructing CISTs has many applications on interconnection networks such as faulttolerant broadcasting and secure message distribution.In underlying graph of communication networks, we want vertices to be close together to avoid communication delays. Pai and Chang [15] provided a unified approach for constructing two CISTs in several hypercube-variant networks, in particular for an n−dimensional hypercube variant network, the diameters of the constructed CISTs were 2n − 1. They asked about the hypercube variant networks which they studied "how to design algorithms to construct more than two CISTs in high dimensional hypercube-variant networks with smaller diameter?"…”