文章基本信息

标题：On the Central Levels Problem
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作者：Petr Gregor ; OndÅej MiÄka ; Torsten M{"u}tze 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：168
页码：60:1-60:17
DOI：10.4230/LIPIcs.ICALP.2020.60
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1-ð" many 1s and at most m+ð" many 1s, i.e., the vertices in the middle 2ð" levels, has a Hamilton cycle for any m â¥ 1 and 1 â¤ ð" â¤ m+1. This problem was raised independently by Savage, by Gregor and Å krekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case ð" = 1, and classical binary Gray codes, namely the case ð" = 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 ð" consecutive levels in the n-dimensional hypercube for any n â¥ 1 and 1 â¤ ð" â¤ n+1. Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the n-dimensional hypercube, nâ¥ 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.
关键词：Gray code; Hamilton cycle; hypercube; middle levels; symmetric chain decomposition