Assume that n and k are positive integers with n ≥ 2k + 1. A non-hamiltonian graph G is hypo hamiltonian if G − v is hamiltonian for any v ∈ V (G). It is proved that the generalized Petersen graph P (n, k) is hypo hamiltonian if and only if k = 2 and n ≡ 5 (mod 6). Similarly, a hamiltonian graph G is hyper hamiltonian if G − v is hamiltonian for any v ∈ V (G). In this paper, we will give some necessary conditions and some sufficient conditions about the hyper hamiltonian generalized Petersen graphs. In particular, P (n, k) is not hyper hamiltonian if n is even and k is odd. We also prove that P (3k, k) is hyper hamiltonian if and only if k is odd. Moreover, P (n, 3) is hyper hamiltonian if and only if n is odd and P (n, 4) is hyper hamiltonian if and only if n = 12. Furthermore, P (n, k) is hyper hamiltonian if k is even with k ≥ 6 and n ≥ 2k + 2 + (4k − 1)(4k + 1), and P (n, k) is hyper hamiltonian if k ≥ 5 is odd and n is odd with n ≥ 6k − 3 + 2k(6k − 2).
Preliminary studies on data mining focus on finding association rules from transaction databases containing items without relationships among them. However, relationships among items often exist in real applications. Most of the previous works only concern about Is-A hierarchy. In this paper, hierarchical relationships include a Has-A hierarchy and multiple Is-A hierarchies are discussed. The proposed method first reduces a Has-A & Is-A hierarchy into an extended Has-A hierarchy using the IsA-Reduce algorithm. The quantitative data is transformed into fuzzy items. The RPFApriori algorithm is then applied to find fuzzy association rules from the fuzzy item data and the extended Has-A hierarchy.
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