On the Vertex Cover Number of 3-Uniform Hypergraph

Given a hypergraph H(V, E), a set of vertices S⊆V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subseteq V$$\end{document} is a vertex cover if every edge has at least one vertex in S. The vertex cover number is the minimum cardinality of a vertex cover, denoted by τ(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (H)$$\end{document}. In this paper, we prove that for every 3-uniform connected hypergraph H(V, E), τ(H)⩽2m+13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (H)\leqslant \frac{2m+1}{3}$$\end{document} 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.