Two-disjoint-cycle-cover bipancyclicity of balanced hypercubes☆
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 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 Qn 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 BHn (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 . 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 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 BHn is bipancyclic. Xu et al. [26] showed that BHn is edge-bipancyclic and Hamiltonian laceable. Cheng et al. [5] showed that BHn admits a two-disjoint-path-cover. Lü and Wu [20] showed that BHn 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 and . 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 . 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 and .
In this paper, we will show that BHn is two-disjoint-cycle-cover -bipancyclic. Since BHn is a bipartite graph and 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 . The definition of the n-dimensional balanced hypercube BHn is defined as follows. Definition 1 (Huang and Wu [23]) For n ≥ 1, an n-dimensional balanced hypercube BHn consists of 22n vertices such that each vertex is labeled by an n-tuple where ai ∈ [3] for and is adjacent to the following 2n vertices: ; for
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 Cℓ and for ℓ ∈ {4, 6, 8}, where Cℓ and 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
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 CQn and locally twisted cubes LTQn are two-disjoint-cycle-cover vertex--pancyclic (resp. edge--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.
References (35)
Pancyclic graphs
J. Combin. Theory Ser. B.
(1971)Cycles embedding in balanced hypercubes with faulty edges and vertices
Discrete Appl. Math.
(2018)Hamiltonian paths and cycles pass through prescribed edges in the balanced hypercubes
Discrete Appl. Math.
(2019)- et al.
Various cycles embedding in faulty balanced hypercubes
Inform. Sci.
(2015) - et al.
Two node-disjoint paths in balanced hypercubes
Appl. Math. Comput.
(2014) - et al.
Vertex-fault-tolerant cycles embedding in balanced hypercubes
Inform. Sci.
(2014) - T.-L. Kung, H.-C. Chen, C.-H. Lin, L.H. Hsu, Three types of two-disjoint-cycle-cover pancyclicity and their...
- et al.
Matching preclusion for balanced hypercubes
Theor. Comput. Sci.
(2012) - et al.
Hamiltonian paths passing through prescribed edges in balanced hypercubes
Theor. Comput. Sci.
(2019) - et al.
Vertex pancyclic graphs
Discrete Appl. Math.
(2002)
Topological properties of hypercubes
IEEE Trans. Comput.
The balanced hypercube: a cube-based system for fault-tolerant applications
IEEE Trans. Comput.
Topological Structure and Analysis of Interconnection Networks
Super connectivity of balanced hypercubes
Appl. Math. Comput.
Conditional diagnosability of balanced hypercubes under the PMC model
Inform. Sci.
Conditional diagnosability of balanced hypercubes under the MM* model
J. Supercomput.
Fault-tolerant-prescribed hamiltonian laceability of balanced hypercubes
Inform. Process. Lett.
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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).