Edge-pancyclicity and path-embeddability of bijective connection graphs

An n-dimensional Bijective Connection graph (in brief BC graph) is a regular graph with 2ⁿ nodes and n2ⁿ⁻¹ edges. The n-dimensional hypercube, crossed cube, Möbius cube, etc. are some examples of the n-dimensional BC graphs. In this paper, we propose a general method to study the edge-pancyclicity and path-embeddability of the BC graphs. First, we prove that a path of length l with dist(X_n, x, y) + 2 ? l ? 2ⁿ − 1 can be embedded between x and y with dilation 1 in X_n for x, y ∈ V(X_n) with x ≠ y in X_n, where X_n (n ? 4) is a n-dimensional BC graph satisfying the three specific conditions and V(X_n) is the node set of X_n. Furthermore, by this result, we can claim that X_n is edge-pancyclic. Lastly, we show that these results can be applied to not only crossed cubes and Möbius cubes, but also other BC graphs except crossed cubes and Möbius cubes. So far, the research on edge-pancyclicity and path-embeddability has been limited in some specific interconnection architectures such as crossed cubes, Möbius cubes.     

Edge-pancyclicity of Möbius cubes

The Möbius cube Mn is a variant of the hypercube Qn and has better properties than Qn with the same number of links and processors. It has been shown by Fan [J. Fan, Hamilton-connectivity and cycle-embedding of Möbius cubes, Inform. Process. Lett. 82 (2002) 113-117] and Huang et al. [W.-T. Huang, W.-K. Chen, C.-H. Chen, Pancyclicity of Möbius cubes, in: Proc. 9th Internat. Conf. on Parallel and Distributed Systems (ICPADS'02), 17-20 Dec. 2002, pp. 591-596], independently, that Mn contains a cycle of every length from 4 to n². In this paper, we improve this result by showing that every edge of Mn lies on a cycle of every length from 4 to n² inclusive.     

The spined cube: A new hypercube variant with smaller diameter

Bijective connection graphs (in brief, BC graphs) are a family of hypercube variants, which contains hypercubes, twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes, etc. It was proved that the smallest diameter of all the known n-dimensional bijective connection graphs (BC graphs) is , given a fixed dimension n. An important question about the smallest diameter among all the BC graphs is: Does there exist a BC graph whose diameter is less than the known BC graphs such as crossed cubes, twisted cubes, Möbius cubes, etc., with the same dimension? This paper answers this question. In this paper, we find that there exists a kind of BC graphs called spined cubes and we prove that the n-dimensional spined cube has the diameter ⌈n/3⌉+3 for any integer n with n?14. It is the first time in literature that a hypercube variant with such a small diameter is presented.     

Embedding a long fault-free cycle in a crossed cube with more faulty nodes

The crossed cube is an important variant of the most popular hypercube network for parallel computing. In this paper, we consider the problem of embedding a long fault-free cycle in a crossed cube with more faulty nodes. We prove that for n?5 and f?2n−7, a fault-free cycle of length at least n²−f−(n−5) can be embedded in an n-dimensional crossed cube with f faulty nodes. Our work extends some previously known results in the sense of the maximum number of faulty nodes tolerable in a crossed cube.     

Parallel construction of independent spanning trees and an application in diagnosis on Möbius cubes

Independent spanning trees (ISTs) on networks have applications to increase fault-tolerance, bandwidth, and security. Möbius cubes are a class of the important variants of hypercubes. A recursive algorithm to construct n ISTs on n-dimensional Möbius cube M _n was proposed in the literature. However, there exists dependency relationship during the construction of ISTs and the time complexity of the algorithm is as high as O(NlogN), where N=2 ⁿ is the number of vertices in M _n and n≥2. In this paper, we study the parallel construction and a diagnostic application of ISTs on Möbius cubes. First, based on a circular permutation n?1,n?2,…,0 and the definitions of dimension-backbone walk and dimension-backbone tree, we propose an O(N) parallel algorithm, called PMCIST, to construct n ISTs rooted at an arbitrary vertex on M _n . Based on algorithm PMCIST, we further present an O(n) parallel algorithm. Then we provide a parallel diagnostic algorithm with high efficiency to diagnose all the vertices in M _n by at most n+1 steps, provided the number of faulty vertices does not exceed n. Finally, we present simulation experiments of ISTs and an application of ISTs in diagnosis on 0-M ₄.     

Edge-fault-tolerant vertex-pancyclicity of augmented cubes

The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube, possesses some properties superior to those of the hypercube. In this paper, we show that every vertex in AQn lies on a fault-free cycle of every length from 3 to n², even if there are up to n−1 edge faults. We also show that our result is optimal.     

Vertex-pancyclicity of twisted cubes with maximal faulty edges

The n-dimensional twisted cube, denoted by TQ _n , a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, we show that every vertex in TQ _n lies on a fault-free cycle of every length from 6 to 2 ⁿ , even if there are up to n?2 link faults. We also show that our result is optimal.     

Embedding two edge-disjoint Hamiltonian cycles into locally twisted cubes

The n-dimensional hypercube network Q_n is one of the most popular interconnection networks since it has simple structure and is easy to implement. The n-dimensional locally twisted cube LTQ_n, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as Q_n. One advantage of LTQ_n is that the diameter is only about half of the diameter of Q_n. Recently, some interesting properties of LTQ_n have been investigated in the literature. The presence of edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the interconnection network. The existence of two edge-disjoint Hamiltonian cycles in locally twisted cubes has remained unknown. In this paper, we prove that the locally twisted cube LTQ_n with n?4 contains two edge-disjoint Hamiltonian cycles. Based on the proof of existence, we further provide an O(n2ⁿ)-linear time algorithm to construct two edge-disjoint Hamiltonian cycles in an n-dimensional locally twisted cube LTQ_n with n?4, where LTQ_n contains 2ⁿ nodes and n2ⁿ⁻¹ edges.     

Paths in Möbius cubes and crossed cubes

The Möbius cube MQn and the crossed cube CQn are two important variants of the hypercube Qn. This paper shows that for any two different vertices u and v in G∈{MQn,CQn} with n?3, there exists a uv-path of every length from dG(u,v)+2 to n²−1 except for a shortest uv-path, where dG(u,v) is the distance between u and v in G. This result improves some known results.     

Fault-tolerant analysis of a class of networks

In this paper, we explore the 2-extraconnectivity of a special class of graphs G(G₀,G₁;M) proposed by Chen et al. [Y.-C. Chen, J.J.M. Tan, L.-H. Hsu, S.-S. Kao, Super-connectivity and super edge-connectivity for some interconnection networks, Applied Mathematics and Computation 140 (2003) 245-254]. As applications of the results, we obtain that the 2-extraconnectivities of several well-known interconnection networks, such as hypercubes, twisted cubes, crossed cubes, Möbius cubes and locally twisted cubes, are all equal to 3n−5 when their dimension n is not less than 8. That is, when n?8, at least 3n−5 vertices must be removed to disconnect any one of these n-dimensional networks provided that the removal of these vertices does not isolate a vertex or an edge.     

Embedding a family of meshes into twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The twisted cube is an important variation of the hypercube. Let TQn denote the n-dimensional twisted cube. In this paper, we consider embedding a family of 2-dimensional meshes into a twisted cube. The main results obtained in this paper are: (1) For any odd integer n?1, there exists a mesh of size 2ײn−1 that can be embedded in the TQn with unit dilation and unit expansion. (2) For any odd integer n?5, there exists a mesh of size 4ײn−2 that can be embedded in the TQn with dilation 2 and unit expansion. (3) For any odd integer n?5, a family of two disjoint meshes of size 4ײn−3 can be embedded into the TQn with unit dilation and unit expansion. Results (1) and (3) are optimal in the sense that the dilations and expansions of the embeddings are unit values.     

Fault-tolerant cycle-embedding of crossed cubes

The crossed cube CQ_n introduced by Efe has many properties similar to those of the popular hypercube. However, the diameter of CQ_n is about one half of that of the hypercube. Failures of links and nodes in an interconnection network are inevitable. Hence, in this paper, we consider the hybrid fault-tolerant capability of the crossed cube. Letting f_e and f_v be the numbers of faulty edges and vertices in CQ_n, we show that a cycle of length l, for any 4?l?|V(CQ_n)|−f_v, can be embedded into a wounded crossed cube as long as the total number of faults (f_v+f_e) is no more than n−2, and we say that CQ_n is (n−2)-fault-tolerant pancyclic. This result is optimal in the sense that if there are n−1 faults, there is no guarantee of having a cycle of a certain length in it.     

Fault-free Hamiltonian cycles in crossed cubes with conditional link faults

The crossed cube, which is a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, assuming that each node is incident with at least two fault-free links, we show that an n-dimensional crossed cube contains a fault-free Hamiltonian cycle, even if there are up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated. We also verify that the assumption is practically meaningful by evaluating its occurrence probability, which is very close to 1.     

Fault-tolerant edge-pancyclicity of locally twisted cubes

The n-dimensional locally twisted cube LTQ_n is a new variant of the hypercube, which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of LTQ_n, and shows that if LTQ_n (n ? 3) contains at most n − 3 faulty vertices and/or edges then, for any fault-free edge e and any integer ? with 6 ? ? ? 2ⁿ − f_v, there is a fault-free cycle of length ? containing the edge e, where f_v is the number of faulty vertices. The result is optimal in some senses. The proof is based on the recursive structure of LTQ_n.     

Node-pancyclicity and edge-pancyclicity of hypercube variants

Twisted cubes, crossed cubes, Möbius cubes, and locally twisted cubes are some of the widely studied hypercube variants. The 4-pancyclicity of twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes and the 4-edge-pancyclicity of twisted cubes, crossed cubes, Möbius cubes are proven in [C.P. Chang, J.N. Wang, L.H. Hsu, Topological properties of twisted cube, Inform. Sci. 113 (1999) 147-167; C.P. Chang, T.Y. Sung, L.H. Hsu, Edge congestion and topological properties of crossed cubes, IEEE Trans. Parall. Distr. 11 (1) (2000) 64-80; J. Fan, Hamilton-connectivity and cycle embedding of the Möbius cubes, Inform. Process. Lett. 82 (2002) 113-117; X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Appl. Math. Lett. 17 (2004) 919-925; J. Fan, N. Yu, X. Jia, X. Lin, Embedding of cycles in twisted cubes with edge-pancyclic, Algorithmica, submitted for publication; J. Fan, X. Lin, X. Jia, Node-pancyclic and edge-pancyclic of crossed cubes, Inform. Process. Lett. 93 (2005) 133-138; M. Xu, J.M. Xu, Edge-pancyclicity of Möbius cubes, Inform. Process. Lett. 96 (2005) 136-140], respectively. It should be noted that 4-edge-pancyclicity implies 4-node-pancyclicity which further implies 4-pancyclicity. In this paper, we outline an approach to prove the 4-edge-pancyclicity of some hypercube variants and we prove in particular that Möbius cubes and locally twisted cubes are 4-edge-pancyclic.     

Efficient unicast in bijective connection networks with the restricted faulty node set

The connectivity is an important criteria to measure the fault-tolerant performance of a graph. However, the connectivity based on the condition of the set of arbitrary faulty nodes is generally lower. In this paper, in order to heighten this measure, we introduce the restricted connectivity into bijective connection networks. First, we prove that the probability that all the neighbors of an arbitrary node becomes faulty in any n-dimensional bijective connection network X_n is very low when n becomes sufficient large. Then, we give a constructive proof that under the condition that each node of an n-dimensional bijective connection network X_n has at least one fault-free neighbor, its restricted connectivity is 2n − 2, about half of the connectivity of X_n. Finally, by our constructive proof, we give an O(n) algorithm to get a reliable path of length at most n + 3⌈log₂∣F∣⌉ + 1 between any two fault-free nodes in an n-dimensional bijective connection network. In particular, since the family of BC networks contains hypercubes, crossed cubes, Möbius cubes, etc., our algorithm is appropriate for these cubes.     

Embedding a family of disjoint multi-dimensional meshes into a crossed cube

Crossed cubes are an important class of hypercube variants. This paper addresses how to embed a family of disjoint multi-dimensional meshes into a crossed cube. We prove that for n?4 and 1?m?⌊n/2⌋−1, a family of m² disjoint k-dimensional meshes of size t₁²×t₂²×?×tk² each can be embedded in an n-dimensional crossed cube with unit dilation, where and max1?i?k{ti}?n−2m−1. This result means that a family of mesh-structured parallel algorithms can be executed on a same crossed cube efficiently and in parallel. Our work extends some recently obtained results.     

Path bipancyclicity of hypercubes

A bipartite graph is vertex-bipancyclic (respectively, edge-bipancyclic) if every vertex (respectively, edge) lies in a cycle of every even length from 4 to |V(G)| inclusive. It is easy to see that every connected edge-bipancyclic graph is vertex-bipancyclic. An n-dimensional hypercube, or n-cube denoted by Qn, is well known as bipartite and one of the most efficient networks for parallel computation. In this paper, we study a stronger bipancyclicity of hypercubes. We prove that every n-dimensional hypercube is (2n−4)-path-bipancyclic for n?3. That is, for any path P of length k with 1?k?2n−4 and any integer l with max{2,k}?l?²n−1, an even cycle C of length 2l can be found in Qn such that the path P is included in C for n?3.     

Highly fault-tolerant cycle embeddings of hypercubes

The hypercube Q_n is one of the most popular networks. In this paper, we first prove that the n-dimensional hypercube is 2n − 5 conditional fault-bipancyclic. That is, an injured hypercube with up to 2n − 5 faulty links has a cycle of length l for every even 4 ⩽ l ⩽ 2ⁿ when each node of the hypercube is incident with at least two healthy links. In addition, if a certain node is incident with less than two healthy links, we show that an injured hypercube contains cycles of all even lengths except hamiltonian cycles with up to 2n − 3 faulty links. Furthermore, the above two results are optimal. In conclusion, we find cycles of all possible lengths in injured hypercubes with up to 2n − 5 faulty links under all possible fault distributions.     

On embedding cycles into faulty twisted cubes

The twisted cube TQ_n is an alternative to the popular hypercube network. Recently, some interesting properties of TQ_n were investigated. In this paper, we study the pancycle problem on faulty twisted cubes. Let f_e and f_v be the numbers of faulty edges and faulty vertices in TQ_n, respectively. We show that, with f_e + f_v ? n − 2, a faulty TQ_n still contains a cycle of length l for every 4 ? l ? ∣V(TQ_n)∣ − f_v and odd integer n ? 3.