A Fibonacci string of order \(n\) is a binary string of length \(n\) with no two consecutive ones. A Fibonacci string of order \(n\) which does not have a one in both the first and last position is called a Lucas string of order \(n\). The Fibonacci cube and the Lucas cube are the subgraphs of the hypercube induced by the set of Fibonacci strings and the set of Lucas strings, respectively. The authors give a definition of an extended Lucas cube and show that all extended Lucas cubes are Hamiltonian with a single exception. It is also shown that the Fibonacci and Lucas cubes and all their extensions contain a Hamming distance path between every pair of their vertices.