A short proof of the middle levels theorem

TL;DR

This paper provides a shorter, more accessible proof of the middle levels conjecture, which states that a specific graph of bitstrings has a Hamilton cycle for all n ≥ 1.

Contribution

The paper introduces a novel, concise proof of the middle levels theorem, simplifying the understanding of this longstanding combinatorial problem.

Findings

Confirmed the existence of Hamilton cycles in the middle levels graph for all n ≥ 1

Provided a more accessible proof compared to previous complex demonstrations

Simplified the combinatorial understanding of the middle levels conjecture

Abstract

Consider the graph that has as vertices all bitstrings of length $2n+1$ with exactly $n$ or $n+1$ entries equal to 1, and an edge between any two bitstrings that differ in exactly one bit. The well-known middle levels conjecture asserts that this graph has a Hamilton cycle for any $n\geq 1$. In this paper we present a new proof of this conjecture, which is much shorter and more accessible than the original proof.