The Fibonacci cube Γ n is a subgraph of n -dimensional hypercube induced by the vertices without two consecutive ones. Klavžar and Žigert Fibonacci cubes are the resonance graphs of fibonaccenes, Fibonacci Quart. 43 (2005) 269–276 proved that Fibonacci cubes are precisely the Z -transformation graphs (or resonance graphs) of zigzag hexagonal chains. In this paper, we characterize plane bipartite graphs whose Z -transformation graphs are exactly Fibonacci cubes. If we delete from Γ n ( n ≥ 3 ) all the vertices with 1 both in the first and in the last position, we obtain the Lucas cube L n . We show, however, that none of the Lucas cubes are Z -transformation graphs, and characterize plane bipartite graphs whose Z -transformation graphs are L 2 k ′ for k ≥ 2 , which is obtained from L 2 k by adding two vertices and joining one to 1010 … 10 and the other to 0101 … 01 .