A universal partial cycle (or upcycle) for $\mathcal{A}^n$ is a cyclic sequence that covers each word of length $n$ over the alphabet $\mathcal{A}$ exactly once -- like a De Bruijn cycle, except that we also allow a wildcard symbol $\mathord{\diamond}$ that can represent any letter of $\mathcal{A}$. Chen et al. in 2017 and Goeckner et al. in 2018 showed that the existence and structure of upcycles are highly constrained, unlike those of De Bruijn cycles, which exist for every alphabet size and word length. Moreover, it was not known whether any upcycles existed for $n \ge 5$. We present several examples of upcycles over both binary and non-binary alphabets for $n = 8$. We generalize two graph-theoretic representations of De Bruijn cycles to upcycles. We then introduce novel approaches to constructing new upcycles from old ones. Notably, given any upcycle for an alphabet of size $a$, we show how to construct an upcycle for an alphabet of size $ak$ for any $k \in \mathbb{N}$, so each example generates an infinite family of upcycles. We also define folds and lifts of upcycles, which relate upcycles with differing densities of $\mathord{\diamond}$ characters. In particular, we show that every upcycle lifts to a De Bruijn cycle. Our constructions rely on a different generalization of De Bruijn cycles known as perfect necklaces, and we introduce several new examples of perfect necklaces. We extend the definitions of certain pseudorandomness properties to partial words and determine which are satisfied by all upcycles, then draw a conclusion about linear feedback shift registers. Finally, we prove new nonexistence results based on the word length $n$, alphabet size, and $\mathord{\diamond}$ density.

翻译：通用部分环（或upcycle）是针对$\mathcal{A}^n$的一种循环序列，它恰好覆盖字母表$\mathcal{A}$上每个长度为$n$的单词一次——类似于德布鲁因环，但允许通配符符号$\mathord{\diamond}$表示$\mathcal{A}$中的任意字母。Chen等人（2017年）和Goeckner等人（2018年）证明，与适用于任意字母表大小和单词长度的德布鲁因环不同，upcycle的存在性和结构受到严格限制。此外，对于$n \ge 5$，此前未知是否存在任何upcycle。我们给出了在二元及非二元字母表上$n = 8$时的多个upcycle示例。我们将德布鲁因环的两种图论表示推广到upcycle。随后，我们引入基于已有upcycle构建新upcycle的新方法。值得注意的是，给定大小为$a$的字母表上的任意upcycle，我们展示了如何为大小为$ak$（$k \in \mathbb{N}$）的字母表构建一个upcycle，因此每个示例都能生成一个无限族upcycle。我们还定义了upcycle的折叠与提升，这些操作关联不同$\mathord{\diamond}$字符密度的upcycle。特别地，我们证明了每个upcycle均可提升为一个德布鲁因环。我们的构造依赖于德布鲁因环的另一种推广——完美项链，并引入了多个新的完美项链示例。我们将某些伪随机性属性的定义扩展至部分词，并确定所有upcycle满足的属性，进而得出关于线性反馈移位寄存器的结论。最后，我们基于单词长度$n$、字母表大小及$\mathord{\diamond}$密度证明了新的不存在性结果。