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      Proof of the middle levels conjecture

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          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.

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          Author and article information

          Journal
          2014-04-17
          2014-08-11
          Article
          1404.4442
          412fa12e-e97b-4852-bd5f-d47cdd2848e8

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          05C45, 94B25
          arXiv admin note: text overlap with arXiv:1111.2413
          math.CO

          Combinatorics
          Combinatorics

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