Hyper-Hamiltonian generalized Petersen graphs

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 (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).

Wirelength of 1-fault hamiltonian graphs into wheels and fans

“…. , n/2 } and generalized Petersen graphs P (n, m) -hypohamiltonian/hyperhamiltonian [1,24,31,32], augmented cubes AQ n -pancyclic [4], crossed cubes CQ n -almost pancyclic [9], Mőbius cubes MQ n -(n − 2)-fault almost pancyclic [7,28], locally twisted cubes LTQ n -almost pancyclic [33], twisted cubes TQ n -(n − 2)-fault almost pancyclic [10,11,16,28], twisted n-cubes T n Q -1-fault hamiltonian [27], generalized twisted cubes GQ n -(n − 2)-fault almost pancyclic [28], hierarchical cubic networks HCN(n) -almost pancyclic [13,15], alternating group graphs AG n -(n − 2)-fault hamiltonian [6], arrangement graphs A n,k -pancyclic [8], tori [19,20]. Further, hypercubes Q n [32], folded hypercubes FQ n [32], shuffle cubes SQ n [22], cube connected cycles CCC(n) [29], cycliccubes G k n [12], wrapped butterfly networks WB(n, k) [17], star graphs S n [18] are hamiltonian.…”

Hyper-Hamiltonian circulants

“…For example, Araki examined hyper-Hamiltonian laceability of Cayley graphs, taking into consideration networks for parallel and distributed systems [1]. Mai et al considered sufficient conditions for a hyper-Hamiltonian cycle in generalized Petersen graphs [8]. Del-Vecchio et al also focused on some sufficient conditions for a graph to be hyper-Hamiltonian by providing spectral and non-spectral conditions for the hyper-Hamiltonian cycle [6].…”

4-ordered-Hamiltonian problems of the generalized Petersen graph GP(n,4)

“…It took considerable effort to finally completely solve the Hamiltonian problem for P(n, k) as given in [18]. The result can actually be extended to the case when an edge or a vertex is deleted from the graph as given in [20]. The corresponding Hamiltonian-connected/laceable problems seem even more difficult as the original conjecture of Alspach is still not settled after over 20 years.…”