## Journal of Applied Mathematics

### The Larger Bound on the Domination Number of Fibonacci Cubes and Lucas Cubes

Shengzhang Ren

#### Abstract

Let ${\mathrm{\Gamma }}_{n}$ and ${\mathrm{\Lambda }}_{n}$ be the $n$-dimensional Fibonacci cube and Lucas cube, respectively. Denote by $\mathrm{\Gamma }[{u}_{n,k,z}]$ the subgraph of ${\mathrm{\Gamma }}_{n}$ induced by the end-vertex ${u}_{n,k,z}$ that has no up-neighbor. In this paper, the number of end-vertices and domination number $\gamma$ of ${\mathrm{\Gamma }}_{n}$ and ${\mathrm{\Lambda }}_{n}$ are studied. The formula of calculating the number of end-vertices is given and it is proved that $\gamma (\mathrm{\Gamma }[{u}_{n,k,z}])\le {2}^{k-1}+1$. Using these results, the larger bound on the domination number $\gamma$ of ${\mathrm{\Gamma }}_{n}$ and ${\mathrm{\Lambda }}_{n}$ is determined.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 954738, 5 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305580

Digital Object Identifier
doi:10.1155/2014/954738

Mathematical Reviews number (MathSciNet)
MR3178979

Zentralblatt MATH identifier
07010806

#### Citation

Ren, Shengzhang. The Larger Bound on the Domination Number of Fibonacci Cubes and Lucas Cubes. J. Appl. Math. 2014 (2014), Article ID 954738, 5 pages. doi:10.1155/2014/954738. https://projecteuclid.org/euclid.jam/1425305580

#### References

• W.-J. Hsu, “Fibonacci cubes. A new interconnection topology,” IEEE Transactions on Parallel and Distributed Systems, vol. 4, no. 1, pp. 3–12, 1993.
• E. Dedó, D. Torri, and N. Z. Salvi, “The observability of the Fibonacci and the Lucas cubes,” Discrete Mathematics, vol. 255, no. 1–3, pp. 55–63, 2002.
• A. Castro, S. Klavžar, M. Mollard, and Y. Rho, “On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2655–2660, 2011.
• A. Castro and M. Mollard, “The eccentricity sequences of Fibonacci and Lucas cubes,” Discrete Mathematics, vol. 312, no. 5, pp. 1025–1037, 2012.
• E. Munarini and N. Z. Salvi, “Structural and enumerative properties of the Fibonacci cubes,” Discrete Mathematics, vol. 255, no. 1–3, pp. 317–324, 2002.
• D. A. Pike and Y. Zou, “The domination number of fibonacci cubes,” Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 80, pp. 433–444, 2012.
• C. Whitehead and N. Z. Salvi, “Observability of the extended Fibonacci cubes,” Discrete Mathematics, vol. 266, no. 1–3, pp. 431–440, 2003.
• J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan, London, UK; Elsevier, New York, NY, USA, 1976. \endinput