Fibonacci-like cubes as -transformation graphs

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 .