Classifying Rotationally-Closed Languages Having Greedy Universal Cycles

TL;DR

This paper classifies all subsets of strings over a finite alphabet that are closed under rotations and admit a greedy universal cycle construction, extending understanding of cyclic structures in combinatorics.

Contribution

It provides a complete classification of rotationally-closed subsets of strings that can be generated by a greedy universal cycle algorithm.

Findings

Identifies all rotationally-closed subsets with greedy universal cycles

Characterizes the structure of such subsets

Extends the theory of universal cycles in combinatorics

Abstract

Let $\textbf{T}(n,k)$ be the set of strings of length $n$ over the alphabet $\Sigma=\{1,2,\ldots,k\}$. A universal cycle for $\textbf{T}(n,k)$ can be constructed using a greedy algorithm: start with the string $k^n$, and continually append the least symbol possible without repeating a substring of length $n$. This construction also creates universal cycles for some subsets $\textbf{S}\subseteq\textbf{T}(n,k)$; we will classify all such subsets that are closed under rotations.