### Abstract

In this paper we generalize the concept of uniquely K
_{r}
-saturated graphs to hypergraphs. Let K
_{r}
^{(k)}
denote the complete k-uniform hypergraph on r vertices. For integers k, r, n such that 2 k < r < n, a k-uniform hypergraph H with n vertices is uniquely K
_{r}
^{(k)}
-saturated if H does not contain K
_{r}
^{(k)}
but adding to H any k-set that is not a hyperedge of H results in exactly one copy of K
_{r}
^{(k)}
. Among uniquely K
_{r}
^{(k)}
-saturated hypergraphs, the interesting ones are the primitive(
_{k−1}
)ones that do not have a dominating vertex—a vertex belonging to all possible (Formula presented) edges. Translating the concept to the complements of these hypergraphs, we obtain a natural restriction of τ-critical hypergraphs: a hypergraph H is uniquely τ-critical if for every edge e, τ (H − e) = τ (H) − 1 and H − e has a unique transversal of size τ (H) − 1. We have two constructions for primitive uniquely K
_{r}
^{(k)}
-saturated hypergraphs. One shows that for k and r where 4 k < r 2k−3, there exists such a hypergraph for every n > r. This is in contrast to the case k = 2 and r = 3 where only the Moore graphs of diameter two have this property. Our other construction keeps n − r fixed; in this case we show that for any fixed k 2 there can only be finitely many examples. We give a range for n where these hypergraphs exist. For n−r = 1 the range is completely determined: k + 1 n (Formula presented). For larger values of n − r the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.

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

Article number | #P4.35 |

Journal | Electronic Journal of Combinatorics |

Volume | 25 |

Issue number | 4 |

Publication status | Published - jan. 1 2018 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

_{r}

^{(K)}-saturated hypergraphs

*Electronic Journal of Combinatorics*,

*25*(4), [#P4.35].

**
Uniquely K
_{r}
^{(K)}
-saturated hypergraphs
.** / Gyárfás, A.; Hartke, Stephen G.; Viss, Charles.

Research output: Article

_{r}

^{(K)}-saturated hypergraphs ',

*Electronic Journal of Combinatorics*, vol. 25, no. 4, #P4.35.

_{r}

^{(K)}-saturated hypergraphs Electronic Journal of Combinatorics. 2018 jan. 1;25(4). #P4.35.

}

TY - JOUR

T1 - Uniquely K r (K) -saturated hypergraphs

AU - Gyárfás, A.

AU - Hartke, Stephen G.

AU - Viss, Charles

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In this paper we generalize the concept of uniquely K r -saturated graphs to hypergraphs. Let K r (k) denote the complete k-uniform hypergraph on r vertices. For integers k, r, n such that 2 k < r < n, a k-uniform hypergraph H with n vertices is uniquely K r (k) -saturated if H does not contain K r (k) but adding to H any k-set that is not a hyperedge of H results in exactly one copy of K r (k) . Among uniquely K r (k) -saturated hypergraphs, the interesting ones are the primitive( k−1 )ones that do not have a dominating vertex—a vertex belonging to all possible (Formula presented) edges. Translating the concept to the complements of these hypergraphs, we obtain a natural restriction of τ-critical hypergraphs: a hypergraph H is uniquely τ-critical if for every edge e, τ (H − e) = τ (H) − 1 and H − e has a unique transversal of size τ (H) − 1. We have two constructions for primitive uniquely K r (k) -saturated hypergraphs. One shows that for k and r where 4 k < r 2k−3, there exists such a hypergraph for every n > r. This is in contrast to the case k = 2 and r = 3 where only the Moore graphs of diameter two have this property. Our other construction keeps n − r fixed; in this case we show that for any fixed k 2 there can only be finitely many examples. We give a range for n where these hypergraphs exist. For n−r = 1 the range is completely determined: k + 1 n (Formula presented). For larger values of n − r the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.

AB - In this paper we generalize the concept of uniquely K r -saturated graphs to hypergraphs. Let K r (k) denote the complete k-uniform hypergraph on r vertices. For integers k, r, n such that 2 k < r < n, a k-uniform hypergraph H with n vertices is uniquely K r (k) -saturated if H does not contain K r (k) but adding to H any k-set that is not a hyperedge of H results in exactly one copy of K r (k) . Among uniquely K r (k) -saturated hypergraphs, the interesting ones are the primitive( k−1 )ones that do not have a dominating vertex—a vertex belonging to all possible (Formula presented) edges. Translating the concept to the complements of these hypergraphs, we obtain a natural restriction of τ-critical hypergraphs: a hypergraph H is uniquely τ-critical if for every edge e, τ (H − e) = τ (H) − 1 and H − e has a unique transversal of size τ (H) − 1. We have two constructions for primitive uniquely K r (k) -saturated hypergraphs. One shows that for k and r where 4 k < r 2k−3, there exists such a hypergraph for every n > r. This is in contrast to the case k = 2 and r = 3 where only the Moore graphs of diameter two have this property. Our other construction keeps n − r fixed; in this case we show that for any fixed k 2 there can only be finitely many examples. We give a range for n where these hypergraphs exist. For n−r = 1 the range is completely determined: k + 1 n (Formula presented). For larger values of n − r the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.

UR - http://www.scopus.com/inward/record.url?scp=85061566027&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061566027&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85061566027

VL - 25

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 4

M1 - #P4.35

ER -