Maximal hypercubes in Fibonacci and Lucas cubes

The Fibonacci cube $Γ_n$ is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1's. The Lucas cube $Λ_n$ is obtained from $Γ_n$ by removing vertices that start and end with 1. We characterize maximal induced hypercubes in $Γ_n$ and $Λ_n$ and deduce for any $p\leq n$ the number of maximal $p$-dimensional hypercubes in these graphs.

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