Niche Number of Linear Hypertrees

TL;DR

This paper investigates the niche number of linear hypertrees with maximum degree two, establishing bounds and constructions for hypertrees with specific niche numbers and degrees.

Contribution

It characterizes the niche number of linear hypertrees with maximum degree two and provides constructions for hypertrees with prescribed maximum degrees and niche number zero.

Findings

If H is a linear hypertree with Δ(H)=2 and anti-rank 3, then its niche number is 0.

The maximum degree condition of 2 is proven to be optimal.

Hypertrees with prescribed maximum degree from 3 to 2r can be constructed with niche number 0.

Abstract

For a digraph $D$, the niche hypergraph $NH(D)$ of $D$ is the hypergraph having the same set of vertices as $D$ and the set of hyperedges is \begin{align} E(NH(D)) &= \{e \subseteq V(D) : |e| \geq 2~and~there~exists~v \in V(D)~such~that~e = N_{D}^{-}(v)\notag &~~~~~~~or~e = N_{D}^{+}(v)\}.\notag \end{align} A digraph is said to be acyclic if it has no directed cycle as a subdigraph. For a given hypergraph $H$, the niche number $\hat{n}(H)$ is the smallest integer such that $H$ together with $\hat{n}(H)$ isolated vertices is the niche hypergraph of an acyclic digraph. In this paper, we study the niche number of linear hypertrees with maximum degree two. By our result, we can conclude for a special case that if $H$ is a linear hypertree with $\Delta(H) = 2$ and anti-rank three, then $\hat{n}(H) = 0$. We also prove that the maximum degree condition is best possible. Moreover, it was proved that if $H$ is a hypergraph of rank $r$ whose niche number is not infinity, then $\Delta(H) \leq 2r$. In this paper, we give a construction of hypertrees whose niche number is $0$ of prescribed maximum degree from $3$ to $2r$.