Fibonacci -cubes as Cartesian products

The Fibonacci ( p , r ) -cube Γ n ( p , r ) 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 substrings of ones. These cubes simultaneously generalize several interconnection networks, notably hypercubes, Fibonacci cubes, and postal networks. In this note it is proved that Γ n ( 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 Γ n ( 2 , 2 ) are non-trivial Cartesian products.