1. On universal partial words
Torsten Mütze
joint work with Herman Chen, Sergey Kitaev and Brian Sun

2. Universal words Given: alphabet , often binary , word length
Definition: A universal word for contains every word over exactly once as a subword of length
Examples:
cyclic setting
linear setting
0011
00110
00, 01, 10, 11
000, 001, 010, 011, 100, 101, 110, 111
length

3. Universal words sometimes called deBruijn sequences
a concept that is many centuries old (sanskrit poetry)
many applications inside and outside of combinatorics
captured in [Knuth TAOCP Vol. 4A 11]
generalizable to other combinatorial structures such as permutations, subsets etc. [Hurlbert 90], [Chung, Diaconis, Graham 92]
Theorem (folklore): For any alphabet and any , there is a cyclic universal word for
Theorem [deBruijn 46]: There are such words.

4. Universal partial words
we now also allow a joker symbol * in addition to letters from
cyclic setting
linear setting
***
000, 001, 010, 011, 100, 101, 110, 111
**0111
**011
111 not covered
**01110
110 covered twice
0*001*11
0*001*110*0
0000, 0100, …, 1111
* *10011
more compact way of representing universal words
generalization of universal words (=no jokers)
words with jokers * are called partial words (large literature)
universal partial word  upword

5. Our results
we initiate the systematic study of universal partial words
no general existence result like before, but also several non-existence results
parameters: alphabet size , word length , number/position of jokers *, cyclic/linear setting

6. Our results – linear setting
single joker *
Thm: If , there is no linear upword with a single *.
binary alphabet , := position of * from the boundary
Thm: There is no binary linear upword if , or if and , or if and
Thm: There is a binary linear upword for any

7. Our results – linear setting
two jokers **,
Thm: A fraction of ways of placing two jokers does not yield a binary linear upword.
Thm: ** and **0111 are the only examples with two adjacent jokers (up to reversal, complements).
Thm: There is an infinite family of binary linear upwords with two jokers *.

8. Our results – cyclic setting
arbitrary number of jokers *
Thm: If , there is no cyclic upword for
Cor: If and is odd, there is no cyclic upword for
Know only a single binary cyclic upword for any : 0*001*11 for (up to rotation, reversal, complements)
Cyclic upwords for any even and have been constructed in a follow-up paper [Goeckner et al. 17+]

9. Proof ideas
Theorem (folklore): For any alphabet and any , there is a cyclic universal word for
Proof: Define the deBruijn graph
Want: a Hamilton cycle in
Observation: is the line graph of
Want: a Eulerian cycle in
is connected and in-degree equals out-degree at every vertex

10. Proof ideas
Consider the binary linear upword 0* for
0*011100
_*______
000
001
011
111
110
100
010
101
Approach: Prove the existence/non-existence of upwords by considering the corresponding subgraphs in (generalizations of Eulerian cycles/paths)

11. Open problems
Existence of binary linear upwords with a single joker at position ? Verified for
More than two jokers? Non-binary alphabets?
Existence of cyclic binary upwords for even ?
Efficient algorithms?

12. Thank you!