Maximal hypercubes in Fibonacci and Lucas cubes

TL;DR

This paper characterizes the largest induced hypercubes within Fibonacci and Lucas cubes, providing formulas for counting maximal hypercubes of any dimension in these graphs.

Contribution

It introduces a complete characterization of maximal hypercubes in Fibonacci and Lucas cubes and derives formulas for their enumeration.

Findings

Characterization of maximal induced hypercubes in Fibonacci and Lucas cubes

Formulas for counting maximal p-dimensional hypercubes

Insights into the structure of Fibonacci and Lucas cubes

Abstract

The Fibonacci cube $\Gamma_n$ is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1's. The Lucas cube $\Lambda_n$ is obtained from $\Gamma_n$ by removing vertices that start and end with 1. We characterize maximal induced hypercubes in $\Gamma_n$ and $\Lambda_n$ and deduce for any $p\leq n$ the number of maximal $p$-dimensional hypercubes in these graphs.