Proof of the middle levels conjecture

Abstract

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length 2 n + 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 ⩾ 1 . This conjecture originated probably with Havel, Buck and Wiedemann, but has also been attributed to Dejter, Erdös, Trotter and various others, and despite considerable efforts, it resisted all attacks during the last 30 years. In this paper, we prove the middle levels conjecture. In fact, we construct 2 2 Ω ( n ) different Hamilton cycles in the middle layer graph, which is best possible.