# Packing parameters in graphs: New bounds and a solution to an open problem

## TL;DR

This paper establishes new bounds on packing parameters in graphs, characterizes extremal graphs, and solves an open problem related to the relationship between packing and clique number.

## Contribution

It introduces improved bounds for packing numbers, characterizes graphs achieving equality, and solves an open problem on connected graphs with specific packing properties.

## Key findings

- Derived upper bounds on packing and open packing numbers.
- Characterized graphs where bounds are tight.
- Solved an open problem on the structure of certain connected graphs.

## Abstract

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].