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Czechoslovak Mathematical Journal

, Volume 69, Issue 1, pp 93–97 | Cite as

A Note on a Conjecture on Niche Hypergraphs

  • Pawaton KaemawichanuratEmail author
  • Thiradet Jiarasuksakun
Article
  • 22 Downloads

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 

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References

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

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  • Pawaton Kaemawichanurat
    • 1
    Email author
  • Thiradet Jiarasuksakun
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceKing Mongkut’s University of Technology ThonburiBangkokThailand

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