Proof of the middle levels conjecture

Abstract

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit. The middle levels conjecture asserts that this graph has a Hamilton cycle for every $n\geq 1$. This conjecture originated probably with Havel, Buck and Wiedemann, but has also been attributed to Dejter, Erd\H{o}s, Trotter and various others, and despite considerable efforts it remained open during the last 30 years. In this paper we prove the middle levels conjecture. In fact, we construct $2^{2^{\Omega(n)}}$ different Hamilton cycles in the middle layer graph, which is best possible.