Extremal hypergraphs for matching number and domination number

TL;DR

This paper generalizes a known inequality relating the domination number and matching number in hypergraphs of any rank, and characterizes the extremal hypergraphs of rank 3 that achieve equality.

Contribution

It extends the inequality 1 1 to all hypergraphs of rank r and fully characterizes extremal hypergraphs of rank 3 where equality holds.

Findings

Established 1 1 for all hypergraphs of rank r.

Characterized extremal hypergraphs of rank 3 with equality.

Generalized previous results from 2-uniform hypergraphs to arbitrary rank.

Abstract

A matching in a hypergraph $\mathcal{H}$ is a set of pairwise disjoint hyperedges. The matching number $\nu(\mathcal{H})$ of $\mathcal{H}$ is the size of a maximum matching in $\mathcal{H}$. A subset $D$ of vertices of $\mathcal{H}$ is a dominating set of $\mathcal{H}$ if for every $v\in V\setminus D$ there exists $u\in D$ such that $u$ and $v$ lie in an hyperedge of $\mathcal{H}$. The cardinality of a minimum dominating set of $\mathcal{H}$ is the domination number of $\mathcal{H}$, denoted by $\gamma(\mathcal{H})$. It was proved that $\gamma(\mathcal{H})\leq (r-1)\nu(\mathcal{H})$ for $r$-uniform hypergraphs and the 2-uniform hypergraphs (graphs) achieving equality $\gamma(\mathcal{H})=\nu(\mathcal{H})$ have been characterized. In this paper we generalize the inequality $\gamma(\mathcal{H})\leq (r-1)\nu(\mathcal{H})$ to arbitrary hypergraph of rank $r$ and we completely characterize the extremal hypergraphs $\mathcal{H}$ of rank $3$ achieving equality $\gamma(\mathcal{H})=(r-1)\nu(\mathcal{H})$.