Maximal hypercubes in Fibonacci and Lucas cubes

The Fibonacci cube ΓnΓn is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube ΛnΛn is obtained from ΓnΓn by removing vertices that start and end with 1. We characterize maximal induced hypercubes in ΓnΓn and ΛnΛn and deduce for any p≤np≤n the number of maximal pp-dimensional hypercubes in these graphs.