Institute of Mathematics

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.