Embedding Fibonacci Cubes into Hypercubes with Omega(2cn) Faulty Nodes

Fibonacci Cubes are special subgraphs of hypercubes based on Fibonacci numbers. We present a construction of a direct embedding of a Fibonacci Cube of dimension n into a faulty hypercube of dimension n with less or equal 2⌈n/4⌉-1 faults. In fact, there exists a direct embedding of a Fibonacci Cube of dimension n into a faulty hypercube of dimension n with at most 2n/4fn faults (fn is the n-th Fibonacci number). Thus the number Φ(n) of tolerable faults grows exponentially with respect to dimension n, Φ(n) = Ω(2cn), for c = 2-log2(1+√5) = 0:31. On the other hand, Φ(n) = O(2dn), for d = (8-3 log2 3)/4 = 0.82. As a corollary, there exists a nearly polynomial algorithm constructing a direct embedding of a Fibonacci Cube of dimension n into a faulty hypercube of dimension n (if it exists) provided that faults are given on input by enumeration. However, the problem is NP-complete, if faults are given on input with an asterisk convention.