Cycles embedding in folded hypercubes under the conditional fault model (original) (raw)
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Let FF e be the set of jFF e j 6 2n À 5 faulty edges in an n-dimensional folded hypercube FQ n such that each vertex of FQ n is incident to at least two fault-free edges, where n P 4 and n is even. Under this assumption, we show that every fault-free edge of FQ n lies on a faultfree cycle of every odd length from n þ 1 to 2 n À 1. In terms of the number of tolerant faulty edges and embedding odd cycles in FQ n , our result improves not only the result in Xu and Ma (2006) where jFF e j ¼ 0, but also the previous best result gotten by Xu et al. (2006) where jFF e j 6 n À 1.
Fault-tolerant cycles embedding in hypercubes with faulty edges
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Let Q n be an n-dimensional hypercube with f e 6 3n À 8 faulty edges and n P 5. In this paper, we consider the faulty hypercube under the following two additional conditions: (1) each vertex is incident to at least two fault-free edges, and (2) every 4-cycle does not have any pair of non-adjacent vertices whose degrees are both two after removing the faulty edges. We prove that there exists a fault-free cycle of every even length from 4 to 2 n in Q n. Our result improves the result by Liu and Wang (2014) in terms of the lengths of embedding cycles, where under the same conditions, a fault-free Hamiltonian cycle was constructed.
Vertex-fault-tolerant cycles embedding in balanced hypercubes
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The balanced hypercube is a new variant of the hypercube, which was proposed by Wu and Huang. Xu et al. (2007) proved that every edge of an n-dimensional balanced hypercube BH n lies on a cycle of every even length from 4 to 2 2n. In this paper, we consider the edge-bipancyclicity of BH n with faulty vertices. Let F v be the set of faulty vertices in BH n with jF v j 6 n À 1. We show that every fault-free edge of BH n À F v lies on a fault-free cycle of every even length from 4 to 2 2n À 2jF v j, where n P 1. Our result improves the previous best result by Xu et al. in terms of fault-tolerant vertices.
Cycles embedding in balanced hypercubes with faulty edges and vertices
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Wu and Huang proposed a new variation of the hypercube, named balanced hypercube, which possesses many good properties such as bipanconnectivity, edge-bipancyclicity, Hamiltonian laceability, hyper Hamiltonian laceability. In this paper, we consider n-dimensional balanced hypercube with |F e | faulty edges and |F v | faulty vertices. We prove that if |F v | + |F e | ≤ n − 1, then every fault-free edge of BH n lies on a fault-free cycle of every even length from 6 to 2 2n − 2|F v |, where n ≥ 2; and if |F v | + |F e | ≤ 2n − 3, then there is a fault-free cycle of every even length from 6 to 2 2n −2|F v | in BH n , where n ≥ 2. Furthermore, we propose the distance between vertex-disjoint edge e and cycle C , i.e., d(e, C) = min{d(e, e ′) | e ′ ∈ E(C)}, where d(e, e ′) =
Fault-tolerant path embedding in folded hypercubes with both node and edge faults
Theoretical Computer Science, 2013
The folded hypercube FQ n is a well-known variation of the hypercube structure. FQ n is superior to Q n in many measurements, such as diameter, fault diameter, connectivity, and so on. LetṼ (FQ n ) (resp.Ẽ(FQ n )) denote the set of faulty nodes (resp. faulty edges) in FQ n . In the case that all nodes in FQ n are fault-free, it has been shown that FQ n contains a fault-free path of length 2 n − 1 (resp. 2 n − 2) between any two nodes of odd (resp. even) distance if |Ẽ(FQ n )| ≤ n − 1, where n ≥ 1 is odd; and FQ n contains a fault-free path of length 2 n − 1 between any two nodes if |Ẽ(FQ n )| ≤ n − 2, where n ≥ 2 is even. In this paper, we extend the above result to obtain two further properties, which consider both node and edge faults, as follows:
Fault-Hamiltonicity of Hypercube-Like Interconnection Networks
19th IEEE International Parallel and Distributed Processing Symposium
We call a graph G to be f-fault hamiltonian (resp. ffault hamiltonian-connected) if there exists a hamiltonian cycle (resp. if each pair of vertices are joined by a hamiltonian path) in G\F for any set F of faulty elements with |F | ≤ f. In this paper, we deal with the graph G 0 ⊕ G 1 obtained from connecting two graphs G 0 and G 1 with n vertices each by n pairwise nonadjacent edges joining vertices in G 0 and vertices in G 1. Provided each G i is ffault hamiltonian-connected and f + 1-fault hamiltonian, 0 ≤ i ≤ 3, we show that G 0 ⊕G 1 is f +1-fault hamiltonianconnected for any f ≥ 2 and f + 2-fault hamiltonian for any f ≥ 1, and that for any f ≥ 0, H 0 ⊕ H 1 is f + 2-fault hamiltonian-connected and f + 3-fault hamiltonian, where
Hamiltonian cycles in hypercubes with faulty edges
Information Sciences an International Journal, 2014
Let F be a set of faulty edges in hypercube Q n with jFj 6 3n À 8 for n P 5. We prove that there still exists a fault-free Hamiltonian cycle in Q n if the following two conditions are satisfied: (1) the degree of every vertex is at least two, and (2) there do not exist a pair of nonadjacent vertices in a 4-cycle whose degrees are both two after faulty edges are removed.
Fault-free Hamiltonian cycles in faulty arrangement graphs
IEEE Transactions on Parallel and Distributed Systems, 1999
The arrangement graph A n,k , which is a generalization of the star graph (n − k = 1), presents more flexibility than the star graph in adjusting the major design parameters: number of nodes, degree, and diameter. Previously, the arrangement graph has proved Hamiltonian. In this paper, we further show that the arrangement graph remains Hamiltonian even if it is faulty. Let F e and F v denote the numbers of edge faults and vertex faults, respectively. We show that A n,k is Hamiltonian when 1) (k = 2 and n − k ≥ 4, or k ≥ 3 and n k k − ≥ + 4 2 ), and F e ≤ k(n − k) − 2, or 2) k ≥ 2, n k k − ≥ + 2
Linearly many faults in 2-tree-generated networks
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In this article we consider a class of Cayley graphs that are generated by certain 3-cycles on the alternating group A n. These graphs are generalizations of the alternating group graph AG n. We look at the case when the 3-cycles form a "tree-like structure," and analyze its fault resiliency. We present a number of structural theorems and prove that even with linearly many vertices deleted, the remaining graph has a large connected component containing almost all vertices.