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# Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words

by Sergey Kitaev 2,* and
1
School of Statistics and Data Science, Nankai University, Tianjin 300071, China
2
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK
3
College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang 830046, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(5), 778; https://doi.org/10.3390/math8050778
Received: 6 April 2020 / Revised: 8 May 2020 / Accepted: 9 May 2020 / Published: 12 May 2020
(This article belongs to the Section Mathematics and Computer Science)
A universal cycle, or u-cycle, for a given set of words is a circular word that contains each word from the set exactly once as a contiguous subword. The celebrated de Bruijn sequences are a particular case of such a u-cycle, where a set in question is the set $A n$ of all words of length n over a k-letter alphabet A. A universal word, or u-word, is a linear, i.e., non-circular, version of the notion of a u-cycle, and it is defined similarly. Removing some words in $A n$ may, or may not, result in a set of words for which u-cycle, or u-word, exists. The goal of this paper is to study the probability of existence of the universal objects in such a situation. We give lower bounds for the probability in general cases, and also derive explicit answers for the case of removing up to two words in $A n$ , or the case when $k = 2$ and $n ≤ 4$ . View Full-Text
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Chen, H.Z.Q.; Kitaev, S.; Sun, B.Y. Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words. Mathematics 2020, 8, 778.

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