Abstract
We prove that if P is a set of at most 2n − 1 edges in a k-ary n-cube, where k ≥ 4 and n ≥ 2, then there is a Hamiltonian cycle on which every edge of P lies if, and only if, the subgraph of the k-ary n-cube induced by the edges of P is a vertex-disjoint collection of paths. This answers a question posed by Wang, Li and Wang who proved the analogous result for 3-ary n-cubes.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ashir, Y.A., Stewart, I.A.: Fault-tolerant embeddings of Hamiltonian circuits in k-ary n-cubes. SIAM Journal on Discrete Mathematics 15(3), 317–328 (2002)
Bose, B., Broeg, B., Kwon, Y., Ashir, Y.: Lee distance and topological properties of k-ary n-cubes. IEEE Transactions on Computers 44(8), 1021–1030 (1995)
Caha, R., Koubek, V.: Hamiltonian cycles and paths with a prescribed set of edges in hypercubes and dense sets. Journal of Graph Theory 51(2), 137–169 (2006)
Chan, M.Y., Lee, S.J.: On the existence of Hamiltonian circuits in faulty hypercubes. SIAM Journal on Discrete Mathematics 4(4), 511–527 (1991)
Dvořák, T.: Hamiltonian cycles with prescribed edges in hypercubes. SIAM Journal on Discrete Mathematics 19(1), 135–144 (2005)
Hsu, L.-H., Lin, C.-K.: Graph Theory and Interconnection Networks. CRC Press, Boca Raton (2009)
Gros, L.: Théorie du Baguenodier, Aimé Vingtrinier, Lyon (1872)
Stewart, I.A., Xiang, Y.: Bipanconnectivity and bipancyclicity in k-ary n-cubes. IEEE Transactions on Parallel and Distributed Systems 20(1), 25–33 (2009)
Wang, S., Li, J., Wang, R.: Hamiltonian cycles with prescribed edges in the 3-ary n-cube. Information Sciences 181(14), 3054–3065 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Stewart, I.A. (2011). Hamiltonian Cycles through Prescribed Edges in k-Ary n-Cubes. In: Wang, W., Zhu, X., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2011. Lecture Notes in Computer Science, vol 6831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22616-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-22616-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22615-1
Online ISBN: 978-3-642-22616-8
eBook Packages: Computer ScienceComputer Science (R0)