The Existence and Structure of Universal Partial Cycles

Abstract

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.