On The Hypercompetition Numbers of Hypergraphs

Abstract

The competition hypergraph $C\mathcal{H}(D)$ of a digraph $D$ is the hypergraph such that the vertex set is the same as $D$ and $e\subseteq V(D)$ is a hyperedge if and only if $e$ contains at least $2$ vertices and $e$ coincides with the in-neighborhood of some vertex $v$ in the digraph $D$. Any hypergraph with sufficiently many isolated vertices is the competition hypergraph of an acyclic digraph. The hypercompetition number $hk(\mathcal{H})$ of a hypergraph $\mathcal{H}$ is defined to be the smallest number of such isolated vertices.

In this paper, we study the hypercompetition numbers of hypergraphs. First, we give two lower bounds for the hypercompetition numbers which hold for any hypergraphs. And then, by using these results, we give the exact hypercompetition numbers for some family of uniform hypergraphs. In particular, we give the exact value of the hypercompetition number of a connected graph.