On the vertex cover number of 3 uniform hypergraph

TL;DR

This paper establishes an upper bound on the vertex cover number for 3-uniform connected hypergraphs, showing it is at most (2m+1)/3, with equality characterizing hypertrees with perfect matchings.

Contribution

It proves a tight bound on the vertex cover number for 3-uniform connected hypergraphs and characterizes the cases of equality.

Findings

Vertex cover number t(H) <= (2m+1)/3 for 3-uniform connected hypergraphs

Equality holds iff H is a hypertree with perfect matching

Provides a characterization of extremal hypergraphs for the bound

Abstract

Given a hypergraph H(V;E), a set of vertices S in V is a vertex cover if every edge has at least a vertex in S. The vertex cover number is the minimum cardinality of a vertex cover, denoted by t(H). In this paper, we prove that for every 3 uniform connected hypergraph H(V;E), t(H)<=(2m+1)/3 holds on where m is the number of edges. Furthermore, the equality holds on if and only if H(V;E) is a hypertree with perfect matching.