Hamiltonian properties on the class of hypercube-like networks

Hamiltonian properties of hypercube variants are explored. Variations of the hypercube networks have been proposed by several researchers. In this paper, we show that all hypercube variants are hamiltonian-connected or hamiltonian-laceable. And we also show that these graphs are bipancyclic.     

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.     

A fault-free Hamiltonian cycle passing through prescribed edges in a hypercube with faulty edges

In this paper, we consider the problem of a fault-free Hamiltonian cycle passing through prescribed edges in an n-dimensional hypercube Qn with some faulty edges. We obtain the following result: Let n?2, F⊂E(Qn), E₀⊂E(Qn)F with 1?|E₀|?2n−3, |F|<n−(⌊|E₀|/2⌋+1). If the subgraph induced by E₀ is a linear forest (i.e., pairwise vertex-disjoint paths), then in the graph Qn−F all edges of E₀ lie on a Hamiltonian cycle.     

Fault-free cycles in folded hypercubes with more faulty elements

Let FFv (respectively, FFe) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional folded hypercube FQn. In this paper, we show that FQn−FFv−FFe contains a fault-free cycle with length at least n²−2|FFv| if |FFe|+|FFv|?2n−4 and |FFe|?n−1, where n?3. Our result improves the previously known result of [S.-Y. Hsieh, A note on cycle embedding in folded hypercubes with faulty elements, Information Processing Letters (2008), in press, doi:10.1016/j.ipl.2008.04.003] where |FFe|+|FFv|?n−1 and n?4.     

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.     

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.     

Generalized honeycomb torus is Hamiltonian

Generalized honeycomb torus is a candidate for interconnection network architectures, which includes honeycomb torus, honeycomb rectangular torus, and honeycomb parallelogramic torus as special cases. Existence of Hamiltonian cycle is a basic requirement for interconnection networks since it helps map a “token ring” parallel algorithm onto the associated network in an efficient way. Cho and Hsu [Inform. Process. Lett. 86 (4) (2003) 185-190] speculated that every generalized honeycomb torus is Hamiltonian. In this paper, we have proved this conjecture.     

On the maximal connected component of hypercube with faulty vertices

In evaluating an interconnection network, it is indispensable to estimate the size of the maximal connected components of the underlying graph when the network begins to lose processors. Hypercube is one of the most popular interconnection networks. This article addresses the maximal connected components of an n-dimensional cube with faulty processors. We first prove that an n-cube with a set F of at most 2n???3 failing processors has a component of size ≥2 ⁿ ???|F|???1. We then prove that an n-cube with a set F of at most 3n???6 missing processors has a component of size ≥2 ⁿ ???|F|???2.     

Strong Menger connectivity with conditional faults on the class of hypercube-like networks

In this paper, we study the Menger property on a class of hypercube-like networks. We show that in all n-dimensional hypercube-like networks with n−2 vertices removed, every pair of unremoved vertices u and v are connected by min{deg(u),deg(v)} vertex-disjoint paths, where deg(u) and deg(v) are the remaining degree of vertices u and v, respectively. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, all hypercube-like networks still have the strong Menger property, even if there are up to 2n−5 vertex faults.     

Fault-free mutually independent Hamiltonian cycles of faulty star graphs

The star graph interconnection network has been recognized as an attractive alternative to the hypercube for its nice topological properties. Unlike previous research concerning the issue of embedding exactly one Hamiltonian cycle into an injured star network, this paper addresses the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty star network. To be precise, let SG _n denote an n-dimensional star network in which f≤n?3 edges may fail accidentally. We show that there exist (n?2?f)-mutually independent Hamiltonian cycles rooted at any vertex in SG _n if n∈{3, 4}, and there exist (n?1?f)-mutually independent Hamiltonian cycles rooted at any vertex in SG _n if n≥5.     

On the maximal connected component of hypercube with faulty vertices (II)

In evaluating the fault tolerance of an interconnection network, it is essential to estimate the size of a maximal connected component of the network at the presence of faulty processors. Hypercube is one of the most popular interconnection networks. In this paper, we prove that for n?≥?6, an n-dimensional cube with a set F of at most (4n???10) failing processors has a component of size ≥2″???|F|???3. This result demonstrates the superiority of hypercube in terms of the fault tolerance.     

Finding cycles in hierarchical hypercube networks

The hierarchical hypercube network, which was proposed as an alternative to the hypercube, is suitable for building a large-scale multiprocessor system. A bipartite graph G=(V,E) is bipancyclic if it contains cycles of all even lengths ranging from 4 to |V|. In this paper, we show that the hierarchical hypercube network is bipancyclic.     

A fault-tolerant wormhole routing scheme for torus networks with nonconvex faults

In this paper, we present a fault-tolerant routing algorithm for torus networks by using only 4 virtual channels. The proposed algorithm is based on the solid fault model, which includes rectangular faults and many practical nonconvex faults. Previous works need at least 6 virtual channels to achieve the same fault-tolerant ability.     

Edge-bipancyclicity of star graphs with faulty elements

In this paper, we investigate the fault-tolerant edge-bipancyclicity of an n-dimensional star graph S_n. Given a set F comprised of faulty vertices and/or edges in S_n with |F|≤n−3 and any fault-free edge e in S_n−F, we show that there exist cycles of every even length from 6 to n!−2|F_v| in S_n−F containing e, where n≥3. Our result is optimal because the star graph is both bipartite and regular with the common degree n−1. The length of the longest fault-free cycle in the bipartite S_n is n!−2|F_v| in the worst case, where all faulty vertices are in the same partite set. We also provide some sufficient conditions from which longer cycles with length from n!−2|F_v|+2 to n!−2|F_v| can be embedded.     

Fault-tolerant wormhole routing for hypercube networks

We present an adaptive fault-tolerant wormhole routing algorithm for hypercubes by using 3 virtual networks. The routing algorithm can tolerate at least n−1 faulty nodes and can route a message via a path of length no more than the shortest path plus four. Previous algorithms which achieve the same fault tolerant ability need 5 virtual networks. Simulation results are also given in this paper.     

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.