On universal partial words

A universal word for a finite alphabet A and some integer n ≥ 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 3 / ∈ A , which can be substituted by any symbol from A. For example, u = 0 3 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 3 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.