# Institute of Mathematics

## A note on a conjecture on niche hypergraphs

###### Received April 21, 2017.   Published online June 21, 2018.

Abstract:  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 $E(N\mathcal{H}(D)) = \{e \subseteq V(D) |e| \geq2$ 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 $\mathcal{C}_m$, $m \geq2$, if $\min\{|e| e \in E(\mathcal{C}_m)\} \geq3$, then $\hat{n}(\mathcal{C}_m) = 0$. In this paper, we prove that this conjecture is true.
Keywords:  niche hypergraph; digraph; linear hypercycle
Classification MSC:  05C65
DOI:  10.21136/CMJ.2018.0182-17

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] C. Garske, M. Sonntag, H. M. Teichert: Niche Hypergraphs. Discuss. Math., Graph Theory 36 (2016), 819-832. DOI 10.7151/dmgt.1893 | MR 3557202 | Zbl 1350.05112

Affiliations:   Pawaton Kaemawichanurat, Thiradet Jiarasuksakun, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand, e-mail: pawaton.kae@kmutt.ac.th, thiradet.jia@kmutt.ac.th

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