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An update on the middle levels problem

Ian Shields·Brendan J. Shields·Carla D. Savage
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Abstract

The middle levels problem is to find a Hamilton cycle in the middle levels, M_{2k+1}, of the Hasse diagram of B_{2k+1} (the partially ordered set of subsets of a 2k+1-element set ordered by inclusion). Previously, the best result was that M_{2k+1} is Hamiltonian for all positive k through k=15. In this note we announce that M_{33} and M_{35} have Hamilton cycles. The result was achieved by an algorithmic improvement that made it possible to find a Hamilton path in a reduced graph of complementary necklace pairs having 129,644,790 vertices, using a 64-bit personal computer.

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Ian Shields, Brendan J. Shields, Carla D. Savage. 2006-08-19. An update on the middle levels problem. https://arxiv.org/abs/math/0608485

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