Covering and 2-degree-packing numbers in graphs

In this paper, we give a relationship between the covering number of a simple graph \(G\), \(\beta(G)\), and a new parameter associated to \(G\), which is called 2-degree-packing number of \(G\), \(\nu_2(G)\). We prove that \[\lceil \nu_{2}(G)/2\rceil\leq\beta(G)\leq\nu_2(G)-1,\] for any simple graph \(G\), with \(|E(G)|>\nu_2(G)\). Also, we give a characterization of connected graphs that attain the equalities.