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

Abstract

Let $p,r$ and $n$ be positive integers. Then the O-Fibonacci $(p,r)$-cube $O{\Gamma }_n^{(p,r)}$ 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 $O{\Gamma }_n^{(p,r)}$ is a non-trivial Cartesian product if and only if $p=1$ and $r\ge n\ge 2$.