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

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.