On Hamiltonian Properties of Honeycomb Meshes

Abstract

In this paper, we investigated Hamiltonian properties of honeycomb meshes which are created in two different ways. We obtained different Hamilton paths for Honeycomb Meshes for any dimension with using n-bit gray code. Finally, we gave an algorithm which is used to label the nodes of Honeycomb Meshes.Interconnection networks are formed of different components of mechanism and connections between them. A network topology is the pattern for connecting one element to other elements and it may vary depending on formation such as tree, bus, mesh, star, ring, hypercube and tori. In this paper, to construct network topology, we use honeycomb meshes using two different structuring, show labeling with gray codes and then analyze Hamilton properties of honeycomb meshes. 

Keywords

• Honeycomb meshes
• Interconnection network
• Hamilton graph
• Gray code
• Perfect Matching

On Hamiltonian Properties of Honeycomb Meshes

Abstract

In this paper, we investigated Hamiltonian properties of honeycomb meshes which are created in two different ways. We obtained different Hamilton paths for Honeycomb Meshes for any dimension with using n-bit gray code. Finally, we gave an algorithm which is used to label the nodes of Honeycomb Meshes.Interconnection networks are formed of different components of mechanism and connections between them. A network topology is the pattern for connecting one element to other elements and it may vary depending on formation such as tree, bus, mesh, star, ring, hypercube and tori. In this paper, to construct network topology, we use honeycomb meshes using two different structuring, show labeling with gray codes and then analyze Hamilton properties of honeycomb meshes. 

Keywords

• Honeycomb meshes
• Interconnection network
• Hamilton graph
• Gray code
• Perfect Matching

References

1. Karci A., Selçuk B. (2014) A new hypercube variant : Fractal Cubic Network Graph. Engineering Science and Technology, An International Journal 18(1): 32-41.
2. Selcuk B., Karcı A. (2017) Connected Cubic Network Graph. Engineering Science and Technology, an International Journal 20(3): 934-943.
3. Zhang Q., Yang X., Li P., Huang G., Feng S., Shen C., Han B., Zhang X., Jin F., Xu F., Lu T. J. (2015). Bioinspired engineering of honeycomb structure - Using nature to inspire human innovation. Progress in Materials Science 74: 332–400.
4. Hales T. C. (2001) The Honeycomb Conjecture. Discrete and Computational Geometry 25(1): 1-22.
5. Lester L. N., Sandor J. (1985) Computer Graphics on a Hexagonal Grid. Comput. Graph 8(4): 401-409.
6. Boudjemai A., Amri R., Mankour A., Salem H., Bouanane M. H., Boutchicha D. (2012) Modal Analysis and Testing of Hexagonal Honeycomb Plates Used for Satellite Structural Design. Mater. Des. 35: 266–275.
7. Engelmary G. C., Cheng M., Bettinger C. J., Borenstein J. T., Langer R., Freed L. E. (2008) Accordion-like Honeycombs for Tissue Engineering of Cardiac Anisotropy. Nature Materials 7: 1003-1010.
8. Carle J., Myoupo J. F., Seme D. (1999) All-to-all Broadcasting Algorithms on Honeycomb Networks and Applications. Parallel Process. Lett. 9(4): 539-550.

9. Manuel P., Rajan B., Rajasingh I., M C. M. (2008) On Minimum Metric Dimension of Honeycomb Networks. J. Discret. Algorithms 6(1): 20-27.
10. Nocetti F. G., Stojmenovic I., Zhang J. (2002) Addressing and Routing in Hexagonal Networks with Applications for Tracking Mobile Users and Connection Rerouting in Cellular Networks. IEEE Trans. Parallel Distrib. Syst. 13(9): 963-971.
11. Rajan B., William A., Grigorious C., Stephen S. (2012) On Certain Topological Indices of Silicate , Honeycomb and Hexagonal Networks. J. Comp. Math. Sci 3(5): 530-535.
12. Lee E. T., Lee M. E. (1999). Algorithms for Generating Generalized Gray Codes. Kybernetes 28(6/7): 837–844.
13. Wilson R. J. (1996). Introduction to Graph Theory (4th ed.). Longman Group Ltd., England.
14. Janson S. (1994). The Numbers of Spanning Trees, Hamilton Cycles and Perfect Matchings in a Random Graph. Combinatorics, Probability and Computing 3(1): 97–126.
15. Derakhshan P., Hussak W. (2013). Star Graph Automorphisms and Disjoint Hamilton Cycles. International Journal of Computer Mathematics 90(3): 483–496.
16. Simonraj F., George A. (2012) Embedding of Poly Honeycomb Networks and the Metric dimension of Star of David Network. International Journal on Applications of Graph Theory in Wireless Ad Hoc Networks and Sensor Networks (GRAPH-HOC) Vol.4, No.4, December.
17. Amutha A., Mary A.A. (2016). Perfect matching and slope number related to honeycomb network. International Journal of Pure and Applied Mathematics, Volume 109 No. 8, 243 – 250.
18. Stojmenovic I., (1997). Honeycomb Networks: Topological Properties and Communication Algorithms, Ieee Transactions on Parallel and Distributed Systems, Vol. 8, No. 10, October.
19. Dong Q., Zhao Q., An Y. (2015) The hamiltonicity of generalized honeycomb torus networks. Information Processing Letters Volume 115, Issue 2, February, Pages 104-111.