{"title":"The Panpositionable Hamiltonicity of k-ary n-cubes","authors":"Chia-Jung Tsai, Shin-Shin Kao","country":null,"institution":"","volume":35,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":889,"pagesEnd":896,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/2528","abstract":"The hypercube Qn is one of the most well-known\r\nand popular interconnection networks and the k-ary n-cube Qk\r\nn is\r\nan enlarged family from Qn that keeps many pleasing properties\r\nfrom hypercubes. In this article, we study the panpositionable\r\nhamiltonicity of Qk\r\nn for k \u2265 3 and n \u2265 2. Let x, y of V (Qk\r\nn)\r\nbe two arbitrary vertices and C be a hamiltonian cycle of Qk\r\nn.\r\nWe use dC(x, y) to denote the distance between x and y on the\r\nhamiltonian cycle C. Define l as an integer satisfying d(x, y) \u2264 l \u2264 1\r\n2 |V (Qk\r\nn)|. We prove the followings:\r\n\u2022 When k = 3 and n \u2265 2, there exists a hamiltonian cycle C\r\nof Qk\r\nn such that dC(x, y) = l.\r\n\u2022 When k \u2265 5 is odd and n \u2265 2, we request that l \/\u2208 S\r\nwhere S is a set of specific integers. Then there exists a\r\nhamiltonian cycle C of Qk\r\nn such that dC(x, y) = l.\r\n\u2022 When k \u2265 4 is even and n \u2265 2, we request l-d(x, y) to be\r\neven. Then there exists a hamiltonian cycle C of Qk\r\nn such\r\nthat dC(x, y) = l.\r\nThe result is optimal since the restrictions on l is due to the\r\nstructure of Qk\r\nn by definition.","references":null,"publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 35, 2009"}