Chia-Jung Tsai and Shin-Shin Kao
The Panpositionable Hamiltonicity of kary ncubes
889 - 895
2009
3
11
International Journal of Mathematical and Computational Sciences
https://publications.waset.org/pdf/2528
https://publications.waset.org/vol/35
World Academy of Science, Engineering and Technology
The hypercube Qn is one of the most wellknown
and popular interconnection networks and the kary ncube Qk
n is
an enlarged family from Qn that keeps many pleasing properties
from hypercubes. In this article, we study the panpositionable
hamiltonicity of Qk
n for k ≥ 3 and n ≥ 2. Let x, y of V (Qk
n)
be two arbitrary vertices and C be a hamiltonian cycle of Qk
n.
We use dC(x, y) to denote the distance between x and y on the
hamiltonian cycle C. Define l as an integer satisfying d(x, y) ≤ l ≤ 1
2 V (Qk
n). We prove the followings
• When k 3 and n ≥ 2, there exists a hamiltonian cycle C
of Qk
n such that dC(x, y) l.
• When k ≥ 5 is odd and n ≥ 2, we request that l ∈ S
where S is a set of specific integers. Then there exists a
hamiltonian cycle C of Qk
n such that dC(x, y) l.
• When k ≥ 4 is even and n ≥ 2, we request ld(x, y) to be
even. Then there exists a hamiltonian cycle C of Qk
n such
that dC(x, y) l.
The result is optimal since the restrictions on l is due to the
structure of Qk
n by definition.
Open Science Index 35, 2009