The larger bound on the domination number of Fibonacci cubes and Lucas cubes (Q1714821)

Summary: Let \(\Gamma_n\) and \(\Lambda_n\) be the \(n\)-dimensional Fibonacci cube and Lucas cube, respectively. Denote by \(\Gamma[u_{n,k,z}]\) the subgraph of \(\Gamma_n\) 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 \(\Gamma_n\) and \(\Lambda_n\) are studied. The formula of calculating the number of end-vertices is given and it is proved that \(\gamma(\Gamma[u_{n,k,z}])\leq 2^{k-1} + 1\). Using these results, the larger bound on the domination number \(\gamma\) of \(\Gamma_n\) and \(\Lambda_n\) is determined.