O-Fibonacci $(p,r)$-cube as Cartesian products

Abstract: Let $p ,r $ and $n $ be positive integers. Then the O-Fibonacci $(p,r)$-cube $O\Gamma^{(p,r)}_{n}$ is the subgraph of $Q_{n}$ induced on the binary words in which there is at least $p-1$ zeros between any two $1$s and there is at most $r$ consecutive $10^{p-1}$. These cubes include a wide range of cubes as their special cases, such as hypercubes, Fibonacci cubes, and postal networks. In this note it is proved that $O\Gamma^{(p,r)}_{n}$ is a non-trivial Cartesian product if and only if $p=1$ and $r\geq n\geq2$.