| Title:
|
A note on a conjecture on niche hypergraphs (English) |
| Author:
|
Kaemawichanurat, Pawaton |
| Author:
|
Jiarasuksakun, Thiradet |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
1 |
| Year:
|
2019 |
| Pages:
|
93-97 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For a digraph $D$, the niche hypergraph $N\mathcal {H}(D)$ of $D$ is the hypergraph having the same set of vertices as $D$ and the set of hyperedges $E(N\mathcal {H}(D)) = \{e \subseteq V(D) \colon |e| \geq 2$ and there exists a vertex $v$ such that $e = N^{-}_{D}(v)$ or $e = N^{+}_{D}(v)\}$. A digraph is said to be acyclic if it has no directed cycle as a subdigraph. For a given hypergraph $\mathcal {H}$, the niche number $\hat {n}(\mathcal {H})$ is the smallest integer such that $\mathcal {H}$ together with $\hat {n}(\mathcal {H})$ isolated vertices is the niche hypergraph of an acyclic digraph. C. Garske, M. Sonntag and H. M. Teichert (2016) conjectured that for a linear hypercycle $\mathcal {C}_{m}$, $m \geq 2$, if $\min \{|e| \colon e \in E(\mathcal {C}_{m})\} \geq 3$, then $\hat {n}(\mathcal {C}_{m}) = 0$. In this paper, we prove that this conjecture is true. (English) |
| Keyword:
|
niche hypergraph |
| Keyword:
|
digraph |
| Keyword:
|
linear hypercycle |
| MSC:
|
05C65 |
| idZBL:
|
Zbl 07088772 |
| idMR:
|
MR3923577 |
| DOI:
|
10.21136/CMJ.2018.0182-17 |
| . |
| Date available:
|
2019-03-08T14:56:51Z |
| Last updated:
|
2021-04-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147620 |
| . |
| Reference:
|
[1] Garske, C., Sonntag, M., Teichert, H. M.: Niche Hypergraphs.Discuss. Math., Graph Theory 36 (2016), 819-832. Zbl 1350.05112, MR 3557202, 10.7151/dmgt.1893 |
| . |
