Fibonacci (p,r)(p,r)-cubes as Cartesian products

The Fibonacci (p,r)(p,r)-cube Γn(p,r) is the subgraph of QnQn induced on binary words of length nn in which there are at most rr consecutive ones and there are at least pp zeros between two substrings of ones. These cubes simultaneously generalize several interconnection networks, notably hypercubes, Fibonacci cubes, and postal networks. In this note it is proved that Γn(p,r) is a non-trivial Cartesian product if and only if p=1p=1 and r=n≥2r=n≥2, or p=r=2p=r=2 and n≥2n≥2, or n=p=3n=p=3 and r=2r=2. This rounds a result from Ou et al. (2011) asserting that Γn(2,2) are non-trivial Cartesian products.