A book proof of the middle levels theorem

TL;DR

This paper presents a concise constructive proof demonstrating the existence of a Hamilton cycle in a specific subgraph of a hypercube, focusing on vertices with exactly n or n+1 ones.

Contribution

It provides a new, short, constructive proof for the middle levels theorem, enhancing understanding of Hamilton cycles in hypercube subgraphs.

Findings

Established a Hamilton cycle in the middle levels graph

Provided a constructive proof approach

Simplified previous proofs of the theorem

Abstract

We give a short constructive proof for the existence of a Hamilton cycle in the subgraph of the $(2n+1)$-dimensional hypercube induced by all vertices with exactly $n$ or $n+1$ many 1s.