Fibonacci-like cubes as Z-transformation graphs

The Fibonacci cube @C"n is a subgraph of n-dimensional hypercube induced by the vertices without two consecutive ones. Klavzar and Zigert [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 @C"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.

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