On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes

Aline Castro Trejo Castro Trejo (1) , Sandi Klavzar (2) , Michel Mollard (1) , Yoomi Rho (3)

1 IF - Institut Fourier
2 FMF - Faculty of Mathematics and Physics [Ljubljana]
3 Department of Mathematics

Let $\Gamma_n$ and $\Lambda_n$ be the $n$-dimensional Fibonacci cube and Lucas cube, respectively. The domination number $\gamma$ of Fibonacci cubes and Lucas cubes is studied. In particular it is proved that $\gamma(\Lambda_{n})$ is bounded below by $\left\lceil\frac{L_{n}-2n}{n-3}\right\rceil$, where $L_n$ is the $n$-th Lucas number. The 2-packing number $\rho$ of these cubes is also studied. It is proved that $\rho(\Gamma_{n})$ is bounded below by $2^{2^{\frac{\lfloor \lg n\rfloor}{2}-1}}$ and the exact values of $\rho(\Gamma_n)$ and $\rho(\Lambda_n)$ are obtained for $n\le 10$. It is also shown that ${\rm Aut}(\Gamma_n) \simeq \mathbb{Z}_2$.

Mots clés: Fibonacci cubes; Lucas cubes; Domination number; 2-packing number; Automorphism group; Computer search

Domaines: Combinatoire [math.CO]

Aline Castro Trejo Castro Trejo, Sandi Klavzar, Michel Mollard, Yoomi Rho. On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes. Computers & Mathematics with Applications, 2011, pp.2655-2660. ⟨hal-00535686v2⟩