  The Fibonacci dimension fdim(G) of a graph G is introduced as the smallest
integer f such that G admits an isometric embedding into Gamma_f, the
f-dimensional Fibonacci cube. We give bounds on the Fibonacci dimension of a
graph in terms of the isometric and lattice dimension, provide a combinatorial
characterization of the Fibonacci dimension using properties of an associated
graph, and establish the Fibonacci dimension for certain families of graphs.
  From the algorithmic point of view we prove that it is NP-complete to decide
if fdim(G) equals to the isometric dimension of G, and that it is also NP-hard
to approximate fdim(G) within (741/740)-epsilon. We also give a
(3/2)-approximation algorithm for fdim(G) in the general case and a
(1+epsilon)-approximation algorithm for simplex graphs.
