On universal partial words

TL;DR

This paper introduces and systematically studies universal partial words, which extend universal words by including joker symbols that can substitute any alphabet letter, exploring their existence, construction, and properties.

Contribution

It initiates the systematic study of universal partial words, providing existence results, explicit constructions, and computational examples.

Findings

Universal partial words can exist with various configurations of joker symbols.

Explicit constructions of universal partial words are provided for different cases.

Computational methods help find examples of universal partial words.

Abstract

A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from $A$ may contain an arbitrary number of occurrences of a special `joker' symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet $A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the subwords $000$ and $010$). We present results on the existence and non-existence of linear and cyclic universal partial words in different situations (depending on the number of $\Diamond$s and their positions), including various explicit constructions. We also provide numerous examples of universal partial words that we found with the help of a computer.