On disjoint hypercubes in Fibonacci cubes

The Fibonacci cube of dimension n, denoted as Γn, is the subgraph of n-cube Qn induced by vertices with no consecutive 1’s. We study the maximum number of disjoint subgraphs in Γn isomorphic to Qk, and denote this number by qk(n). We prove several recursive results for qk(n), in particular we prove that qk(n) = qk−1(n − 2) + qk(n − 3). We also prove a closed formula in which qk(n) is given in terms of Fibonacci numbers, and finally we give the generating function for the sequence {qk(n)}n=0.