### Abstract

of a computer.

Original language | English |
---|---|

Article number | 16 |

Number of pages | 19 |

Journal | Discrete Mathematics and Theoretical Computer Science |

Volume | 19 |

Issue number | 1 |

Publication status | Accepted/In press - 5 May 2017 |

### Fingerprint

### Keywords

- universal word
- partial word
- de Bruijn graph
- Eulerian cycle
- Hamiltonian cycle

### Cite this

*Discrete Mathematics and Theoretical Computer Science*,

*19*(1), [16].

}

*Discrete Mathematics and Theoretical Computer Science*, vol. 19, no. 1, 16.

**On universal partial words.** / Chen, Herman Z.Q.; Kitaev, Sergey; Mütze, Torsten; Sun, Brian Y.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On universal partial words

AU - Chen, Herman Z.Q.

AU - Kitaev, Sergey

AU - Mütze, Torsten

AU - Sun, Brian Y.

PY - 2017/5/5

Y1 - 2017/5/5

N2 - 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 helpof a computer.

AB - 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 helpof a computer.

KW - universal word

KW - partial word

KW - de Bruijn graph

KW - Eulerian cycle

KW - Hamiltonian cycle

UR - http://dmtcs.episciences.org/

M3 - Article

VL - 19

JO - Discrete Mathematics and Theoretical Computer Science

JF - Discrete Mathematics and Theoretical Computer Science

SN - 1365-8050

IS - 1

M1 - 16

ER -