On Disjoint hypercubes in Fibonacci cubes

TL;DR

This paper investigates the maximum number of disjoint subgraphs isomorphic to hypercubes within Fibonacci cubes, providing recursive formulas, closed-form solutions involving Fibonacci numbers, and generating functions.

Contribution

It introduces recursive relations, closed-form formulas, and generating functions for counting disjoint hypercube subgraphs in Fibonacci cubes, advancing combinatorial understanding.

Findings

Recursive formula for q_k(n): q_k(n) = q_{k-1}(n-2) + q_k(n-3)

Closed-form expression for q_k(n) using Fibonacci numbers

Derived generating function for the sequence q_k(n)

Abstract

The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in $\Gamma\_n$ isomorphic to $Q\_k$, and denote this number by $q\_k(n)$. We prove several recursive results for $q\_k(n)$, in particular we prove that $q\_{k}(n) = q\_{k-1}(n-2) + q\_{k}(n-3)$. We also prove a closed formula in which $q\_k(n)$ is given in terms of Fibonacci numbers, and finally we give the generating function for the sequence $\{q\_{k}(n)\}\_{n=0}^{ \infty}$.