On Disjoint hypercubes in Fibonacci cubes

The {\em Fibonacci cube} of dimension $n$, denoted as $\mathrm{\Gamma }\mathrm{\_}n$, is the subgraph of $n$-cube $Q\mathrm{\_}n$ induced by vertices with no consecutive 1’s. We study the maximum number of disjoint subgraphs in $\mathrm{\Gamma }\mathrm{\_}n$ isomorphic to $Q\mathrm{\_}k$, and denote this number by $q\mathrm{\_}k(n)$. We prove several recursive results for $q\mathrm{\_}k(n)$, in particular we prove that $q_{k}(n) = q_{k-1}(n-2) + q_{k}(n-3)$\mathrm{.}Wealsoproveaclosedformulainwhich$q_k(n)$ is given in terms of Fibonacci numbers, and finally we give the generating function for the sequence $\{q\mathrm{\_}k(n)\}\mathrm{\_}{n=0}^{\mathrm{\infty }}$.