Abstract
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 .
Highlights
A universal cycle, or u-cycle, for a given set S withwords of length n over an alphabet A is a circular word u0 u1 · · · u−1 that contains each word from S exactly once as a contiguous subword ui ui+1 · · · ui+n−1 for some 0 ≤ i ≤ ` − 1, where the indices are taken modulo
The celebrated de Bruijn sequences are a particular case of such a u-cycle, where a set in question is the set An of all words of length n over a k-letter alphabet A
A universal word, or u-word, for S is a word u0 u1 · · · u+n−2 that contains each word from S exactly once as a contiguous subword ui ui+1 · · · ui+n−1 for some
Summary
A universal cycle, or u-cycle, for a given set S withwords of length n over an alphabet A is a circular word u0 u1 · · · u−1 that contains each word from S exactly once (and no other word) as a contiguous subword ui ui+1 · · · ui+n−1 for some 0 ≤ i ≤ ` − 1, where the indices are taken modulo. To justify (2) we note that if all of the s removed edges come from the binary cycles considered above, the same lower bound as in (1) will be obtained.
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