de Bruijn序列的一个无向变分
An Unoriented Variation on de Bruijn Sequences
摘要：
对于正整数$k,n$，de Bruijn序列$B(k,n)$是从$k$字符中提取的元素的有限序列，其长度为$n$的子字恰好是$k$字符上长度为$n$的$k^n$字。本文介绍了无方向de Bruijn序列$uB(k,n)$，一个类似于de Bruijn序列，但该序列是向前和向后读取，以确定长度为$n$的子字集。证明了最优长度的非平凡无向de Bruijn序列存在的充要条件是$k$为2或奇数，且$n$小于或等于3.任意$k$，$n$的无向de Bruijn序列都可以由无向de Bruijn图的欧拉化中的某些欧拉路来构造.
英文摘要：
For positive integers $k,n$, a de Bruijn sequence $B(k,n)$ is a finite sequence of elements drawn from $k$ characters whose subwords of length $n$ are exactly the $k^n$ words of length $n$ on $k$ characters. This paper introduces the unoriented de Bruijn sequence $uB(k,n)$, an analog to de Bruijn sequences, but for which the sequence is read both forwards and backwards to determine the set of subwords of length $n$. We show that nontrivial unoriented de Bruijn sequences of optimal length exist if and only if $k$ is two or odd and $n$ is less than or equal to 3. Unoriented de Bruijn sequences for any $k$, $n$ may be constructed from certain Eulerian paths in Eulerizations of unoriented de Bruijn graphs.
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