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

TL;DR

This paper introduces the O-Fibonacci $(p,r)$-cube, a generalized graph structure encompassing various known cubes, and characterizes when it can be decomposed into a Cartesian product.

Contribution

It provides a necessary and sufficient condition for the O-Fibonacci $(p,r)$-cube to be a non-trivial Cartesian product, expanding understanding of its structural properties.

Findings

O-Fibonacci $(p,r)$-cube includes hypercubes, Fibonacci cubes, and postal networks as special cases.

The cube is a non-trivial Cartesian product if and only if $p=1$ and $r \\geq n \\geq 2$.

The paper characterizes the structural conditions for Cartesian product decomposition.

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 $r\geq n\geq2$.