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The Panpositionable Hamiltonicity of k-ary n-cubes
Abstract:The hypercube Qn is one of the most well-known and popular interconnection networks and the k-ary n-cube 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 l-d(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.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1057403Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1231
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