Fibonacci-like cubes as Z-transformation graphs

The Fibonacci cube Γn 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 Γn(n≥3) all the vertices with 1 both in the first and in the last position, we obtain the Lucas cube Ln. We show, however, that none of the Lucas cubes are Z-transformation graphs, and characterize plane bipartite graphs whose Z-transformation graphs are L2k′ for k≥2, which is obtained from L2k by adding two vertices and joining one to 1010…10 and the other to 0101…01.