Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, August 20, 2020, 12:15 pm

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Torsten Mütze (University of Warwick)

On the central levels problem

The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1-k many 1s and at most m+k many 1s, i.e., the vertices in the middle 2k levels, has a Hamilton cycle for any m≥1 and 1≤k≤m+1. This problem was raised independently by Buck and Wiedemann, Savage, Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case k=1, and classical binary Gray codes, namely the case k=m+1.
In this talk I present a general constructive solution of the central levels problem. Our results also imply the existence of optimal cycles through any sequence of k consecutive levels in the n-dimensional hypercube for any n≥1 and 1≤k≤n+1. Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the n-dimensional hypercube, n≥2, that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code (here is a link to experiment with this algorithm: http://combos.org/chains).

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