Non covered vertices in Fibonacci cubes by a maximum set of disjoint hypercubes

TL;DR

This paper proves that asymptotically all vertices in Fibonacci cubes are covered by a maximum set of disjoint hypercubes, solving an open problem about vertex coverage in these graphs.

Contribution

It establishes that asymptotically, the entire Fibonacci cube can be covered by disjoint hypercubes, advancing understanding of their structure.

Findings

Almost all vertices are covered by disjoint hypercubes asymptotically.

Answers an open problem on vertex coverage in Fibonacci cubes.

Provides structural insights into Fibonacci cube tilings.

Abstract

The 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. In this short note we prove that asymptotically all vertices of $\Gamma$ n are covered by a maximum set of disjoint subgraphs isomorphic to Q k , answering an open problem proposed in [2].