The Fibonacci hypercube

The Fibonacci Hypercube is defined as the polytope determined by the convex hull of the “Fibonacci” strings, i.e., binary strings of length n having no consecutive ones. We obtain an efficient characterization of vertex adjacency and use this to study the graph of the Fibonacci Hypercube. In particular we discuss a decomposition of the graph into self-similar subgraphs that are also graphs of Fibonacci hypercubes of lower dimension, we obtain vertex degrees, a recurrence formula for the number of edges, show that the graph is Hamiltonian and study some additional connectivity properties. We conclude with some related open problems.