The Fibonacci (p,r)-cube Γn(p,r) is the subgraph of Qn 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 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=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 Γn(2,2) are non-trivial Cartesian products.
Klavžar, S., & Rho, Y. (2014). Fibonacci (p,r)-cubes as Cartesian products. Discrete Mathematics, 328, 23–26. https://doi.org/10.1016/j.disc.2014.03.027