  In this paper, we investigate the packing parameters in graphs. By applying
the Mantel's theorem, We give upper bounds on packing and open packing numbers
of triangle-free graphs along with characterizing the graphs for which the
equalities hold and exhibit sharp Nordhaus-Gaddum type inequalities for packing
numbers. We also solve the open problem of characterizing all connected graphs
with $\rho_{o}(G)=n-\omega(G)$ posed in [S. Hamid and S. Saravanakumar, {\em
Packing parameters in graphs}, Discuss Math. Graph Theory, 35 (2015), 5--16].
