On universal partial words

1. On universal partial words Torsten Mütze jointworkwithHerman Chen, SergeyKitaev and Brian Sun

2. Universal words • Given:alphabet , oftenbinary ,wordlength • Definition: A universal wordforcontainseverywordoverexactlyonce as a subword of length • Examples: cyclicsetting linearsetting • 0011 • 00, 01, 10, 11 • 00110 • 000, 001, 010, 011, 100, 101, 110, 111 • 00010111 • 0001011100 length

3. Universal words • sometimescalleddeBruijnsequences • a conceptthatismanycenturiesold (sanskritpoetry) • manyapplicationsinside and outside of combinatorics • captured in [Knuth TAOCP Vol. 4A 11] • generalizable to othercombinatorialstructures such as permutations, subsets etc. [Hurlbert 90], [Chung, Diaconis, Graham 92] • Theorem (folklore): For anyalphabet and any ,thereis a cyclic universal wordfor . • Theorem [deBruijn 46]: Thereare such words.

4. Universal partial words • wenow also allow a jokersymbol* in addition to lettersfrom cyclicsetting linearsetting • 000, 001, 010, 011, 100, 101, 110, 111 • *** • **0111 • **011 111 not covered • **01110 110 coveredtwice • 0*001*11 • 0*001*110*0 • 0000, 0100, …, 1111 • *0001011*10011 • morecompact way of representing universal words • generalization of universal words (=no jokers) • wordswithjokers * arecalledpartial words(large literature) • universal partial word upword

5. Ourresults • weinitiate the systematicstudy of universal partial words • no generalexistenceresultlikebefore, but also severalnon-existenceresults • parameters: alphabetsize , wordlength , number/position of jokers *, cyclic/linear setting

6. Ourresults – linear setting • singlejoker * • Thm:If , thereis no linear upwordwith a single *. • binaryalphabet , := position of * from the boundary • Thm:Thereis no binary linearupwordif ,orif and ,orif and . • Thm:Thereis a binary linearupwordforany .

7. Ourresults – linear setting • twojokers **, • Thm: A -fraction of ways ofplacingtwojokersdoes not yield abinary linear upword. • Thm: ** and **0111 are the onlyexampleswithtwoadjacentjokers (up to reversal, complements). • Thm:Thereis an infinite family ofbinary linear upwordswithtwojokers *.

8. Ourresults – cyclicsetting • arbitrarynumber of jokers * • Thm:If , thereis no cyclicupwordfor . • Cor:If and isodd, thereis no cyclicupwordfor . • Knowonly a singlebinarycyclicupwordforany :0*001*11 for (up to rotation, reversal, complements) • Cyclicupwordsforanyeven and havebeenconstructed in a follow-uppaper[Goeckner et al. 17+]

9. Proofideas • Theorem (folklore): For anyalphabet and any ,thereis a cyclic universal wordfor . • Proof: Define the deBruijngraph Want: a Hamilton cycle in Observation: is the line graph of Want: a Euleriancycle in isconnected and in-degreeequalsout-degree at everyvertex

10. Proofideas Consider the binary linear upword 0*011100 for 0*011100 _*______ 000 001 011 111 110 100 101 010 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 upwordswith a singlejoker at position ? Verifiedfor . • Morethantwojokers?Non-binaryalphabets? • Existence of cyclicbinaryupwordsforeven ? • Efficientalgorithms?

12. Thankyou!