# On the probability of existence of a universal cycle or a universal word for a set of words

## TL;DR

This paper investigates the likelihood of universal cycles and words existing for sets of words formed by removing some words from all words of a fixed length over an alphabet, providing bounds and explicit cases.

## Contribution

It introduces probabilistic bounds for the existence of universal cycles and words when subsets are formed by removing words, including specific cases for small parameters.

## Key findings

- Lower bounds for the probability of existence in general cases
- Explicit probability calculations for removing up to two words
- Results for binary alphabet with word length up to four

## 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 $n\leq 4$.