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 of length n appears exactly once as a (consecutive) subword. 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 ◊  A, which can be substituted by any symbol from A. For example, u = 0◊011100 is a universal 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 several result on the existence and non-existence of universal partial words in different situations (depending on the number of ◊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.