# Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2**

## 2. The Case of the Alphabet of Size $K\ge 3$

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**4.**

**Proof.**

- remove any non-loop edge among the $k(k-3)$ edges coming not from special two-cycles, and the remaining edges can be removed from loops. This gives $k(k-3)\left(\genfrac{}{}{0pt}{}{k}{s-1}\right)$ possibilities;
- remove the k edges in either the cycle $1\to 2\to \cdots \to k\to 1$, or the cycle $k\to (k-1)\to \cdots \to k\to 1$, and then remove one more non-loop edge among the $k(k-3)$ edges coming not from special two-cycles, and the remaining edges can be removed from loops, which gives $2k(k-3)\left(\genfrac{}{}{0pt}{}{k}{s-k-1}\right)$ possibilities.

## 3. The Case of the Alphabet Size $\mathit{k}=\mathbf{2}$

**Proof.**

## 4. Exact Values of ${\mathit{P}}_{\mathit{c}}(\mathit{n},\mathit{k},\mathbf{2})$ and ${\mathit{P}}_{\mathit{w}}(\mathit{n},\mathit{k},\mathbf{2})$

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

- either a or b is a loop, in which case clearly a Hamiltonian path in ${B}^{\prime}(n,k)$ exists, or
- $a\to b$ (or $b\to a$) is an edge in $B(n,k)$, in which case a Hamiltonian path in ${B}^{\prime}(n,k)$ exists by Theorem 7 with one exception.

## 5. Directions of Further Research

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Chung, F.; Diaconis, P.W.; Graham, R.L. Universal cycles for combinatorial structures. Discrete Math.
**1992**, 110, 43–59. [Google Scholar] [CrossRef] [Green Version] - Gardner, K.B.; Godbole, A. Universal cycles of restricted words. J. Combin. Math. Combin. Comput.
**2018**, 106, 153–173. [Google Scholar] - Moreno, E. De Bruijn sequences and De Bruijn graphs for a general language. Inform. Process. Lett.
**2005**, 96, 214–219. [Google Scholar] [CrossRef] - Zerbino, D.; Birney, E. Velvet: Algorithms for de novo short read assembly using de Bruijn graphs. Genome Res.
**2008**, 18, 821–829. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Baumslag, M. An algebraic analysis of the connectivity of DeBruijn and shuffle-exchange digraphs. Discrete Appl. Math.
**1995**, 61, 213–227. [Google Scholar] [CrossRef] [Green Version] - Johnson, J.R. Universal cycles for permutations. Discrete Math.
**2009**, 309, 5264–5270. [Google Scholar] [CrossRef] [Green Version] - Kitaev, S.; Potapov, V.N.; Vajnovszki, V. On shortening u-cycles and u-words for permutations. Discrete Appl. Math.
**2019**, in press. [Google Scholar] [CrossRef] [Green Version] - Casteels, K.; Stevens, B. Universal cycles of (n − 1)-partitions of an n-set. Discrete Math.
**2009**, 309, 5332–5340. [Google Scholar] [CrossRef] [Green Version] - Jackson, B. Universal cycles of k-subsets and k-permutations. Discrete Math.
**1993**, 117, 114–150. [Google Scholar] [CrossRef] - Hurlbert, G.; Johnson, T.; Zahl, J. On universal cycles for multisets. Discrete Math.
**2009**, 309, 5321–5327. [Google Scholar] [CrossRef] [Green Version] - Brockman, G.; Kay, B.; Snively, E. On universal cycles of labeled graphs. Electron. J. Combin.
**2010**, 17, 1–9. [Google Scholar] [CrossRef] [Green Version] - Wong, D. Novel Universal Cycle Constructions for a Variety of Combinatorial Objects. Ph.D. Thesis, University of Guelph, Guelph, ON, Canada, 2015. [Google Scholar]
- Sridhar, M.A.; Raghavendra, C.S. Fault-tolerant networks based on the de Bruijn graph. IEEE Trans. Comput.
**1991**, 40, 1167–1174. [Google Scholar] [CrossRef]

s | 1 | 2 | 3 |
---|---|---|---|

${P}_{c}(2,2,s)$ | $\frac{1}{2}$ | $\frac{1}{6}$ | 0 |

${P}_{w}(2,2,s)$ | 1 | $\frac{5}{6}$ | 1 |

s | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

${P}_{c}(3,2,s)$ | $\frac{1}{4}$ | $\frac{1}{14}$ | $\frac{1}{28}$ | $\frac{3}{70}$ | $\frac{1}{28}$ | 0 | 0 |

${P}_{w}(3,2,s)$ | 1 | $\frac{5}{7}$ | $\frac{13}{28}$ | $\frac{5}{14}$ | $\frac{5}{14}$ | $\frac{13}{28}$ | 1 |

s | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${P}_{c}$ | $\frac{1}{8}$ | $\frac{1}{60}$ | $\frac{1}{140}$ | $\frac{3}{910}$ | $\frac{1}{546}$ | $\frac{1}{728}$ | $\frac{1}{1144}$ | $\frac{1}{1287}$ | $\frac{1}{1430}$ | $\frac{1}{1144}$ | $\frac{1}{728}$ | $\frac{3}{1820}$ | $\frac{1}{280}$ | 0 | 0 |

${P}_{w}$ | 1 | $\frac{13}{30}$ | $\frac{13}{70}$ | $\frac{1}{10}$ | $\frac{23}{364}$ | $\frac{355}{8008}$ | $\frac{199}{5720}$ | $\frac{62}{2145}$ | $\frac{153}{5720}$ | $\frac{31}{1144}$ | $\frac{3}{91}$ | $\frac{1}{20}$ | $\frac{13}{140}$ | $\frac{29}{120}$ | 1 |

**Table 4.**References for the lower bounds for ${P}_{c}(n,k,s)$ and ${P}_{w}(n,k,s)$. The gray cells refer to exact values.

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**MDPI and ACS Style**

Chen, H.Z.Q.; Kitaev, S.; Sun, B.Y.
Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words. *Mathematics* **2020**, *8*, 778.
https://doi.org/10.3390/math8050778

**AMA Style**

Chen HZQ, Kitaev S, Sun BY.
Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words. *Mathematics*. 2020; 8(5):778.
https://doi.org/10.3390/math8050778

**Chicago/Turabian Style**

Chen, Herman Z. Q., Sergey Kitaev, and Brian Y. Sun.
2020. "Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words" *Mathematics* 8, no. 5: 778.
https://doi.org/10.3390/math8050778