The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1-𝓁 many 1s and at most m+𝓁 many 1s, i.e., the vertices in the middle 2𝓁 levels, has a Hamilton cycle for any m ≥ 1 and 1 ≤ 𝓁 ≤ m+1. This problem was raised independently by Savage, by Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case 𝓁 = 1, and classical binary Gray codes, namely the case 𝓁 = m+1. In this paper we present a general constructive solution of the central levels problem. Our results also imply the existence of optimal cycles through any sequence of 𝓁 consecutive levels in the n-dimensional hypercube for any n ≥ 1 and 1 ≤ 𝓁 ≤ 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.
@InProceedings{gregor_et_al:LIPIcs.ICALP.2020.60, author = {Gregor, Petr and Mi\v{c}ka, Ond\v{r}ej and M\"{u}tze, Torsten}, title = {{On the Central Levels Problem}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {60:1--60:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.60}, URN = {urn:nbn:de:0030-drops-124678}, doi = {10.4230/LIPIcs.ICALP.2020.60}, annote = {Keywords: Gray code, Hamilton cycle, hypercube, middle levels, symmetric chain decomposition} }
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