Two-disjoint-cycle-cover bipancyclicity of balanced hypercubes

doi:10.1016/j.amc.2020.125305

Applied Mathematics and Computation

Volume 381, 15 September 2020, 125305

https://doi.org/10.1016/j.amc.2020.125305 Get rights and content

Highlights

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BH_n is two-disjoint-cycle-cover [4, 2 $^{2 n - 1}$ ]-bipancyclic.
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This result is optimal in the sense of length ℓ in [4, 2 $^{2 n - 1}$ ].
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This result improves the known results about Hamiltonicity of BH_n.
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This result improves the bipancyclicity of BH_n.

Abstract

A bipartite graph G is two-disjoint-cycle-cover [r₁, r₂]-bipancyclic if for any even integer ℓ satisfying r₁ ≤ ℓ ≤ r₂, there exist two vertex-disjoint cycles C₁ and C₂ in G such that $| V (C_{1}) | = ℓ$ and $| V (C_{2}) | = | V (G) | - ℓ,$ where |V(G)| denotes the number of vertices in G. In this paper, we study the two-disjoint-cycle-cover bipancyclicity of the n-dimensional balanced hypercube BH_n, which is a hypercube-variant network and is superior to hypercube due to having a smaller diameter. As a consequence, we show that BH_n is two-disjoint-cycle-cover $[4, 2^{2 n - 1}]$ -bipancyclic for n ≥ 2.

Introduction

It is well known that a massive multiprocessor system has important applications in parallel and distributed computing. Usually, such a system is modeled by a simple connected graph $G = (V (G), E (G)),$ where V(G) represents the set of processors and E(G) represents the set of communication links between processors. Among all kinds of network topologies, the n-dimensional hypercube Q_n is one of the most popular due to its rich topological properties [22]. However, a potential drawback of the hypercube is its larger diameter compared with others. To overcome such a drawback, numerous hypercube-variant networks have been put forward. One such network topology is the balanced hypercube BH_n (defined later in the next section), which was first proposed by Wu and Huang [23] in 1997 and possesses superior properties to improve efficiency.

For balanced hypercubes, many basic properties have been obtained, such as bipartite [23], vertex-transitivity [23], edge-transitivity [35], (conditional) matching preclusion number [18], diverse (edge) connectivities [7], [16], [17], [28], [29], [35], and diverse diagnosabilities in distinct models [10], [27], [30], [31], [33]. In particular, many properties related to Hamiltonicity have been investigated in recent years. To introduce some results, we need the following notations and graph terminology.

For positive integers i, j with i < j, let $[i, j] = {i, i + 1, \dots, j}$ . A graph G is bipartite if V(G) can be divided into two disjoint sets V₀ and V₁ (where {V₀, V₁} is called a bipartition) such that every edge of G is with one end in V₀ and the other end in V₁. A graph H is a spanning subgraph of G (or equivalently, H spans G) if $V (H) = V (G)$ and E(H)⊆E(G). For u, v ∈ V(G), a path joining u and v is called a (u, v)-path(or (v, u)-path). A cycle (resp. path) that contains every vertex of a graph exactly once is called a Hamiltonian cycle (resp. Hamiltonian path). A graph G is Hamiltonian if it possesses a Hamiltonian cycle, and is Hamiltonian-connected if there is a Hamiltonian (u, v)-path for any two distinct vertices u, v ∈ V(G). A bipartite graph with bipartition {V₀, V₁} is said to be Hamiltonian laceable if for arbitrary vertices u ∈ V₀ and v ∈ V₁, there is a Hamiltonian (u, v)-path.

Bondy [1] defined that a graph G is pancyclic if it contains a cycle of length ℓ (i.e., an ℓ-cycle) for every ℓ ∈ [3, |V(G)|]. More specifically, Randerath et al. [21] defined that a graph G is vertex-pancyclic (resp. edge-pancyclic) if every vertex (resp. edge) of G lies on an ℓ-cycle for every ℓ ∈ [3, |V(G)|]. Similarly, a bipartite graph G is bipancyclic if it contains an ℓ-cycle for every even integer ℓ ∈ [4, |V(G)|]. Furthermore, with the consideration that every vertex and/or edge lies on each even-length cycle, we can define a bipartite graph to be vertex-bipancyclic and/or edge-bipancyclic, respectively. Since cycle (resp. path) embedding is a fundamental issue in evaluating the capability of interconnection networks [12], [24], [25], a network with the above various types of pancyclicity can embed one cycle of every possible length. The research results of cycle (resp. path) embedding in balanced hypercubes have a great deal of wealthy. Wu and Huang [23] first showed that BH_n is bipancyclic. Xu et al. [26] showed that BH_n is edge-bipancyclic and Hamiltonian laceable. Cheng et al. [5] showed that BH_n admits a two-disjoint-path-cover. Lü and Wu [20] showed that BH_n for n ≥ 2 possesses two edge-disjoint Hamiltonian cycles. For the results of Hamiltonian cycle passing through a set of prescribed edges, the reader can refer to [3], [19], [32]. Moreover, for fault-tolerant cycle (resp. path) embedding with distinct conditions, please also refer to [2], [4], [6], [8], [9], [11], [15], [32], [34].

In very recently, Kung and Chen [13] concerned about the problem of embedding more than one cycle that spans all vertices in a graph simultaneously. A two-disjoint-cycle-cover of a graph G is a collection of two vertex-disjoint cycles of G that just spans G. Extending this concept together with pancyclicity, a graph G is two-disjoint-cycle-cover [r₁, r₂]-pancyclic if for any integer ℓ satisfying ℓ ∈ [r₁, r₂], there exist two vertex-disjoint cycles C₁ and C₂ in G such that $| C_{1} | = ℓ$ and $| C_{2} | = | V (G) | - ℓ$ . In this case, we say that {C₁, C₂} is a two-disjoint-cycle-cover of G. From the above definition, it can be seen that $r_{2} \leq \frac{| V (G) |}{2}$ . Then, the problem of two-disjoint-cycle-cover pancyclicity was studied by Kung and Chen [13] for crossed cubes, and by Kung et al. [14] for locally twisted cubes. In addition, Dirac-type and Ore-type sufficient conditions for graphs admitting two-disjoint-cycle-cover pancyclicity were provided in Kung and Chen [13] and Kung et al. [14], respectively. Similarly, a bipartite graph G is two-disjoint-cycle-cover [r₁, r₂]-bipancyclic if for any even integer ℓ satisfying ℓ ∈ [r₁, r₂], there exist two vertex-disjoint cycles C₁ and C₂ in G such that $| C_{1} | = ℓ$ and $| C_{2} | = | V (G) | - ℓ$ .

In this paper, we will show that BH_n is two-disjoint-cycle-cover $[4, 2^{2 n - 1}]$ -bipancyclic. Since BH_n is a bipartite graph and $| V (B H_{n}) | = 2^{2 n},$ this result is best possible.

The remaining parts of this paper are organized as follows. Section 2 gives some necessary definitions and preliminary results. Section 3 proves our main result. Finally, a concluding remark is given in the last section.

Section snippets

Preliminaries

Let n be a positive integer and $[n] = {0, 1, \dots, n}$ . The definition of the n-dimensional balanced hypercube BH_n is defined as follows.

Definition 1

(Huang and Wu [23]) For n ≥ 1, an n-dimensional balanced hypercube BH_n consists of 2²ⁿ vertices such that each vertex is labeled by an n-tuple $(a_{0}, a_{1}, \dots, a_{n - 1}),$ where a_i ∈ [3] for $i \in [n - 1],$ and is adjacent to the following 2n vertices:

(1)
$(a_{0} \pm 1, a_{1}, \dots, a_{n - 1})$ ;
(2)
$(a_{0} \pm 1, a_{1}, \dots, a_{j - 1}, a_{j} + {(- 1)}^{a_{0}}, a_{j + 1}, \dots, a_{n - 1})$ for $j \in [n - 1] ∖ {0},$

where arithmetics in the above elements are taken modulo 4.

From

Main results

From Fig. 2(a)–(c), it is easy to check that BH₂ contains two vertex-disjoint cycles C_ℓ and $C_{16 - ℓ}$ for ℓ ∈ {4, 6, 8}, where C_ℓ and $C_{16 - ℓ}$ are represented by bold lines and dashed lines, respectively. Thus, we have the following result.

Theorem 1

Let BH₂ be the 2-dimensional balanced hypercube. Then BH₂ is two-disjoint-cycle-cover [4,8]-bipancyclic.

Lemma 3

Let BH₃ be the 3-dimensional balanced hypercube. Then BH₃ is two-disjoint-cycle-cover [4,10]-bipancyclic.

Proof

We show that there is a two-disjoint-cycle-cover ${C_{ℓ}, C_{64 - ℓ}$

Conclusion

So far the study of two-disjoint-cycle-cover pancyclicity for networks has received less attention, except for the results of crossed cubes [13] and locally twisted cubes [14]. To obtain the results, the authors in Kung and Chen [13], Kung et al. [14] indeed provided a more stronger proof by induction which showed that crossed cubes CQ_n and locally twisted cubes LTQ_n are two-disjoint-cycle-cover vertex- $[4, 2^{n - 1}]$ -pancyclic (resp. edge- $[6, 2^{n - 1}]$ -pancyclic) for n ≥ 5, i.e., all cycles with possible

Acknowledgment

The authors express their sincere thanks to the anonymous referees for their valuable suggestions which greatly improved the original manuscript.

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^☆: This work was supported by the National Natural Science Foundation of China under the projects 11971054, 11731002 and the 111 Project of China B16002 (R.-X. Hao) and by the Ministry of Science and Technology of Taiwan under the grant MOST-107-2221-E-141-001-MY3 (J.-M. Chang).

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Two-disjoint-cycle-cover bipancyclicity of balanced hypercubes☆

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Main results

Conclusion

Acknowledgment

J. Combin. Theory Ser. B.

Discrete Appl. Math.

Discrete Appl. Math.

Inform. Sci.

Appl. Math. Comput.

Inform. Sci.

Theor. Comput. Sci.

Theor. Comput. Sci.

Discrete Appl. Math.

IEEE Trans. Comput.

IEEE Trans. Comput.

Appl. Math. Comput.

Inform. Sci.

J. Supercomput.

Inform. Process. Lett.