A note on a conjecture on niche hypergraphs

Kaemawichanurat, Pawaton, and Jiarasuksakun, Thiradet. "A note on a conjecture on niche hypergraphs." Czechoslovak Mathematical Journal 69.1 (2019): 93-97. <http://eudml.org/doc/294202>.

@article{Kaemawichanurat2019,
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 $E(N\mathcal \{H\}(D)) = \lbrace e \subseteq V(D) \colon |e| \ge 2$ and there exists a vertex $v$ such that $e = N^\{-\}_\{D\}(v)$ or $e = N^\{+\}_\{D\}(v)\rbrace $. 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 $\mathcal \{C\}_\{m\}$, $m \ge 2$, if $\min \lbrace |e| \colon e \in E(\mathcal \{C\}_\{m\})\rbrace \ge 3$, then $\hat\{n\}(\mathcal \{C\}_\{m\}) = 0$. In this paper, we prove that this conjecture is true.},
author = {Kaemawichanurat, Pawaton, Jiarasuksakun, Thiradet},
journal = {Czechoslovak Mathematical Journal},
keywords = {niche hypergraph; digraph; linear hypercycle},
language = {eng},
number = {1},
pages = {93-97},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on a conjecture on niche hypergraphs},
url = {http://eudml.org/doc/294202},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Kaemawichanurat, Pawaton
AU - Jiarasuksakun, Thiradet
TI - A note on a conjecture on niche hypergraphs
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 1
SP - 93
EP - 97
AB - 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 $E(N\mathcal {H}(D)) = \lbrace e \subseteq V(D) \colon |e| \ge 2$ and there exists a vertex $v$ such that $e = N^{-}_{D}(v)$ or $e = N^{+}_{D}(v)\rbrace $. 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 $\mathcal {C}_{m}$, $m \ge 2$, if $\min \lbrace |e| \colon e \in E(\mathcal {C}_{m})\rbrace \ge 3$, then $\hat{n}(\mathcal {C}_{m}) = 0$. In this paper, we prove that this conjecture is true.
LA - eng
KW - niche hypergraph; digraph; linear hypercycle
UR - http://eudml.org/doc/294202
ER -