Fibonacci (p, r)-cubes as Cartesian products

The Fibonacci (p, r)-cube Gamma((p,r))(n) is the subgraph of Q(n) induced on binary words of length n in which there are at most r consecutive ones and there are at least p zeros between two sub-strings of ones. These cubes simultaneously generalize several interconnection networks, notably hypercubes, Fibonacci cubes, and postal networks. In this note it is proved that Gamma((p,r))(n) is a non-trivial Cartesian product if and only if p = 1 and r = n >= 2, or p = r = 2 and n >= 2, or n = p = 3 and r = 2. This rounds a result from Ou et al. (2011) asserting that Gamma((2.2))(n) are non-trivial Cartesian products. (C) 2014 Elsevier B.V. All rights reserved.