# A Note on a Conjecture on Niche Hypergraphs

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## 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 \(E(N\mathcal{H}(D)) = \{ e \subseteq V(D):|e| \geqslant 2\) and there exists a vertex *v* such that \(e = \mathop N\nolimits_D^ - (v)\) or \(\left. {e = {\rm{ }}N_D^ + (v)} \right\}\). 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 \(\mathcal{C}_m,\;m\geqslant2\), if \(\min \left\{ {\left| e \right|:e \in E({\mathcal{C}_m})} \right\} \geqslant 3\), then \(\hat n(\mathcal{C}_m)=0\). In this paper, we prove that this conjecture is true.

## Keywords

niche hypergraph digraph linear hypercycle## MSC 2010

05C65## Preview

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## References

- [1]
*C. Garske, M. Sonntag, H. M. Teichert*: Niche Hypergraphs, Discuss. Math., Graph Theory.*36*(2016), 819–832.MathSciNetCrossRefzbMATHGoogle Scholar