Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 93-97, 2019
A note on a conjecture on niche hypergraphs
Pawaton Kaemawichanurat, Thiradet Jiarasuksakun
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
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