On some super fault-tolerant Hamiltonian graphs

Chen, Y-Chuang; Tsai, Chang-Hsiung; Hsu, Lih‐Hsing; Tan, Jinglu

doi:10.1016/s0096-3003(02)00933-5

Cited by 30 publications

(18 citation statements)

References 12 publications

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“…Combining this fact with the hypohamiltonicity of G and Γ, we obtain that in G Research impulses from fault-tolerant designs in computer networks-for faulttolerance problems in graph theory see Hayes' paper [65] and, for more recent developments, [21,73,104]-led to considering P1 in various lattices. Such embeddability problems (also concerning longest cycles) are beyond the scope of this Dissertation, but an overview can be found in [114].…”

Section: Empty Intersection Of All Longest Pathsmentioning

confidence: 91%

On Hypohamiltonian and Almost Hypohamiltonian Graphs

Zamfirescu

2014

Journal of Graph Theory

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A graph G is almost hypohamiltonian if G is non‐hamiltonian, there exists a vertex w such that G−w is non‐hamiltonian, and for any vertex v≠w the graph G−v is hamiltonian. We prove the existence of an almost hypohamiltonian graph with 17 vertices and of a planar such graph with 39 vertices. Moreover, we find a 4‐connected almost hypohamiltonian graph, while Thomassen's question whether 4‐connected hypohamiltonian graphs exist remains open. We construct planar almost hypohamiltonian graphs of order n for every n≥74. During our investigation we draw connections between hypotraceable, hypohamiltonian, and almost hypohamiltonian graphs, and discuss a natural extension of almost hypohamiltonicity. Finally, we give a short argument disproving a conjecture of Chvátal (originally disproved by Thomassen), strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems.

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Section: Empty Intersection Of All Longest Pathsmentioning

confidence: 91%

On Hypohamiltonian and Almost Hypohamiltonian Graphs

Zamfirescu

2014

Journal of Graph Theory

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“…Proof We prove this theorem by inducting on n. It is proved in [6,7] that for n = 2 and n = 3, the result holds for G (1..2) and G (1..3) . So G (1..2) is (k −2+1)-Hamiltonian and (k − 3 + 1)-Hamiltonian connected for k ≥ 5.…”

Section: Theorem 1 Formentioning

confidence: 96%

“…Twisted-cubes, crossed-cubes, Möbius cubes, and recursive circulant graphs are proved to be optimal fault-tolerant Hamiltonian and optimal faulttolerant Hamiltonian connected [6,7,[18][19][20]26]. All these families of graphs have some good properties in common, including that they can all be recursively constructed.…”

Section: Introductionmentioning

confidence: 99%

A family of Hamiltonian and Hamiltonian connected graphs with fault tolerance

et al. 2009

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Processor (vertex) faults and link (edge) faults may happen when a network is used, and it is meaningful to consider networks (graphs) with faulty processors and/or links. A k-regular Hamiltonian and Hamiltonian connected graph G is optimal fault-tolerant Hamiltonian and Hamiltonian connected if G remains Hamiltonian after removing at most k − 2 vertices and/or edges and remains Hamiltonian connected after removing at most k − 3 vertices and/or edges. In this paper, we investigate in constructing optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected graphs. Therefore, some of the generalized hypercubes, twistedcubes, crossed-cubes, and Möbius cubes are optimal fault-tolerant Hamiltonian and optimal fault-tolerant Hamiltonian connected.

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“…In the case where both faulty vertices and faulty edges are considered, Huang et al [90] and Chen et al [29] showed, independently, that CQ n is (n − 2)-fault-tolerant hamiltonian for n 3. Yang et al [161] improved this by showing the following result.…”

Section: Theorem 35 (Fu [56]) F Q N Contains a Fault-free Cycle Withmentioning

confidence: 99%

Survey on path and cycle embedding in some networks

2009

Front. Math. China

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To find a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To find cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.

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On some super fault-tolerant Hamiltonian graphs

Cited by 30 publications

References 12 publications

On Hypohamiltonian and Almost Hypohamiltonian Graphs

On Hypohamiltonian and Almost Hypohamiltonian Graphs

A family of Hamiltonian and Hamiltonian connected graphs with fault tolerance

Survey on path and cycle embedding in some networks

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