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.     

An efficient fault-tolerant routing algorithm in bijective connection networks with restricted faulty edges

In this paper, we study fault-tolerant routing in bijective connection networks with restricted faulty edges. First, we prove that the probability that all the incident edges of an arbitrary node become faulty in an n-dimensional bijective connection network, denoted by X_n, is extremely low when n becomes sufficient large. Then, we give an O(n) algorithm to find a fault-free path of length at most n+3⌈log₂∣F∣⌉+1 between any two different nodes in X_n if each node of X_n has at least one fault-free incident edge and the number of faulty edges is not more than 2n−3. In fact, we also for the first time provide an upper bound of the fault diameter of all the bijective connection networks with the restricted faulty edges. Since the family of BC networks contains hypercubes, crossed cubes, Möbius cubes, etc., all the results are appropriate for these cubes.     

Optimal fault-tolerant Hamiltonicity of star graphs with conditional edge faults

The star graph is viewed as an attractive alternative to the hypercube. In this paper, we investigate the Hamiltonicity of an n-dimensional star graph. We show that for any n-dimensional star graph (n≥4) with at most 3n−10 faulty edges in which each node is incident with at least two fault-free edges, there exists a fault-free Hamiltonian cycle. Our result improves on the previously best known result for the case where the number of tolerable faulty edges is bounded by 2n−7. We also demonstrate that our result is optimal with respect to the worst case scenario, where every other node of a cycle of length 6 is incident with exactly n−3 faulty noncycle edges.     

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.     

Pancyclicity and bipancyclicity of conditional faulty folded hypercubes

A graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive; and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order. The pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network’s topology can simulate rings of various lengths. An n-dimensional folded hypercube FQ_n is an attractive variant of an n-dimensional hypercube Q_n that is obtained by establishing some extra edges between the vertices of Q_n. FQ_n for any odd n is known to be bipartite. In this paper, we explore the pancyclicity and bipancyclicity of FQ_n. For any FQ_n (n ? 2) with at most 2n − 3 faulty edges, where each vertex is incident to at least two fault-free edges, we prove that there exists a fault-free cycle of every even length from 4 to 2ⁿ; and when n ? 2 is even, we prove there also exists a fault-free cycle of every odd length from n + 1 to 2ⁿ − 1. The result is optimal with respect to the number of faulty edges tolerated.     

A (4n − 9)/3 diagnosis algorithm on n-dimensional cube network

As a generalization of the precise and pessimistic diagnosis strategies of system-level diagnosis of multicomputers, the t/k diagnosis strategy can significantly improve the self-diagnosing capability of a system at the expense of no more than k fault-free processors (nodes) being mistakenly diagnosed as faulty. In the case k ? 2, to our knowledge, there is no known t/k diagnosis algorithm for general diagnosable system or for any specific system. Hypercube is a popular topology for interconnecting processors of multicomputers. It is known that an n-dimensional cube is (4n − 9)/3-diagnosable. This paper addresses the (4n − 9)/3 diagnosis of n-dimensional cube. By exploring the relationship between a largest connected component of the 0-test subgraph of a faulty hypercube and the distribution of the faulty nodes over the network, the fault diagnosis of an n-dimensional cube can be reduced to those of two constituent (n − 1)-dimensional cubes. On this basis, a diagnosis algorithm is presented. Given that there are no more than 4n − 9 faulty nodes, this algorithm can isolate all faulty nodes to within a set in which at most three nodes are fault-free. The proposed algorithm can operate in O(N log₂ N) time, where N = 2ⁿ is the total number of nodes of the hypercube. The work of this paper provides insight into developing efficient t/k diagnosis algorithms for larger k value and for other types of interconnection networks.     

One-to-one communication in twisted cubes under restricted connectivity

The dimensions of twisted cubes are only limited to odd integers. In this paper, we first extend the dimensions of twisted cubes to all positive integers. Then, we introduce the concept of the restricted faulty set into twisted cubes. We further prove that under the condition that each node of the n-dimensional twisted cube TQ _n has at least one fault-free neighbor, its restricted connectivity is 2n − 2, which is almost twice as that of TQ _n under the condition of arbitrary faulty nodes, the same as that of the n-dimensional hypercube. Moreover, we provide an O(NlogN) fault-free unicast algorithm and simulations result of the expected length of the fault-free path obtained by our algorithm, where N denotes the node number of TQ _n . Finally, we propose a polynomial algorithm to check whether the faulty node set satisfies the condition that each node of the n-dimensional twisted cube TQ _n has at least one fault-free neighbor.     

Pancyclicity of ternary n-cube networks under the conditional fault model

A graph G is said to be conditional k-edge-fault pancyclic if after removing k faulty edges from G, under the assumption that each vertex is incident to at least two fault-free edges, the resulting graph contains a cycle of every length from its girth to |V(G)|. In this paper, we consider ternary n-cube networks and show that they are conditional (4n−5)-edge-fault pancyclic.     

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.     

Long paths and cycles in hypercubes with faulty vertices

A fault-free path in the n-dimensional hypercube Q_n with f faulty vertices is said to be long if it has length at least 2ⁿ-2f-2. Similarly, a fault-free cycle in Q_n is long if it has length at least 2ⁿ-2f. If all faulty vertices are from the same bipartite class of Q_n, such length is the best possible. We show that for every set of at most 2n-4 faulty vertices in Q_n and every two fault-free vertices u and v satisfying a simple necessary condition on neighbors of u and v, there exists a long fault-free path between u and v. This number of faulty vertices is tight and improves the previously known results. Furthermore, we show for every set of at most n²/10+n/2+1 faulty vertices in Q_n where n?15 that Q_n has a long fault-free cycle. This is a first quadratic bound, which is known to be asymptotically optimal.     

Conditional fault-tolerant hamiltonicity of star graphs

The star graph possesses many nice topological properties. In this study, we show that for any n-dimensional star graph (n ⩾ 4) with ⩽2n − 7 edge faults in which each node is incident to at least two non-faulty edges, there exists a fault-free Hamiltonian cycle. Compared with the corresponding study in hypercube, our method is rather succinct. Additionally, we also show the probability that an n dimensional star graph with arbitrary 2n − 7 faulty edges at most is Hamiltonian is very close to one.     

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 pancyclicity of augmented cubes

As an enhancement on the hypercube Qn, the augmented cube AQn, prosed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], not only retains some favorable properties of Qn but also possesses some embedding properties that Qn does not. For example, AQn is pancyclic, that is, AQn contains cycles of arbitrary length for n?2. This paper shows that AQn remains pancyclic provided faulty vertices and/or edges do not exceed 2n−3 and n?4.     

Many-to-many disjoint paths in faulty hypercubes

This paper considers the problem of many-to-many disjoint paths in the hypercube Q_n with f_v faulty vertices and f_e faulty edges, and obtains the following result. For any integer k with 1?k?n-1, any two sets S and T of k fault-free vertices in different parts, if f_v+f_e?n-k-1, then there exist k disjoint fault-free (S,T)-paths in Q_n which contains at least 2ⁿ-2f_v vertices. This result is optimal in the worst case.     

Edge-fault-tolerant bipanconnectivity of hypercubes

This paper shows that for any two distinct vertices u and v with distance d in the hypercube Q_n (n?3) with at most faulty edges and each vertex incident with least two fault-free edges, there exist fault-free uv-paths of length ? in Q_n for every ? with d+4???2ⁿ-1 and . This result improves some known results on edge-fault bipanconnectivity of hypercubes. The proof is based on the recursive structure of Q_n.     

Edge-bipancyclicity of the k-ary n-cubes with faulty nodes and edges

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. In this paper, we investigate the edge-bipancyclicity of k-ary n-cubes with faulty nodes and edges. It is proved that every healthy edge of the faulty k-ary n-cube with f_v faulty nodes and f_e faulty edges lies in a fault-free cycle of every even length from 4 to kⁿ − 2f_v (resp. kⁿ − f_v) if k ? 4 is even (resp. k ? 3 is odd) and f_v + f_e ? 2n − 3. The results are optimal with respect to the number of node and edge faults tolerated.     

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.     

Cycles embedding in hypercubes with node failures

The hypercube has been widely used as the interconnection network in parallel computers. The n-dimensional hypercube Qn is a graph having n² vertices each labeled with a distinct n-bit binary strings. Two vertices are linked by an edge if and only if their addresses differ exactly in the one bit position. Let fv denote the number of faulty vertices in Qn. For n?3, in this paper, we prove that every fault-free edge and fault-free vertex of Qn lies on a fault-free cycle of every even length from 4 to n²−2fv inclusive even if fv?n−2. Our results are optimal.     

The super connectivity of augmented cubes

The augmented cube AQn, proposed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a (2n−1)-regular (2n−1)-connected graph (n≠3). This paper determines that the super connectivity of AQn is 4n−8 for n?6 and the super edge-connectivity is 4n−4 for n?5. That is, for n?6 (respectively, n?5), at least 4n−8 vertices (respectively, 4n−4 edges) of AQn are removed to get a disconnected graph that contains no isolated vertices. When the augmented cube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.     

Long paths in hypercubes with conditional node-faults

Let F be a set of f?2n-5 faulty nodes in an n-cube Q_n such that every node of Q_n still has at least two fault-free neighbors. Then we show that Q_n-F contains a path of length at least 2ⁿ-2f-1 (respectively, 2ⁿ-2f-2) between any two nodes of odd (respectively, even) distance. Since the n-cube is bipartite, the path of length 2ⁿ-2f-1 (or 2ⁿ-2f-2) turns out to be the longest if all faulty nodes belong to the same partite set. As a contribution, our study improves upon the previous result presented by [J.-S. Fu, Longest fault-free paths in hypercubes with vertex faults, Information Sciences 176 (2006) 759-771] where only n-2 faulty nodes are considered.