Hyper-Hamiltonian generalized Petersen graphs

  • Authors:

  • Ta-Cheng Mai;Jeng-Jung Wang;Lih-Hsing Hsu

  • Affiliations:

  • Department of Information Engineering, I-Shou University, Kaohsiung, 84008 Taiwan, ROC;Department of Information Engineering, I-Shou University, Kaohsiung, 84008 Taiwan, ROC;Department of Computer Science and Information Engineering, Providence University, Taichung 43301, Taiwan, ROC

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2008

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Abstract

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(mod6). 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 for 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 n12. 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).