On the packing numbers in graphs

TL;DR

This paper investigates upper bounds on open packing and k-limited packing numbers in graphs, characterizes specific graph classes, and improves existing bounds by relating packing numbers to double domination.

Contribution

It provides new bounds on packing numbers, characterizes graphs with specific packing properties, and links packing numbers to double domination to refine previous bounds.

Findings

Characterized all connected graphs with open packing number equal to n/δ(G).

Established bounds on open packing and 2-limited packing numbers.

Improved existing upper bounds by relating packing numbers to double domination.

Abstract

In this paper, we find upper bounds on the open packing and $k$-limited packing numbers with emphasis on the cases $k=1$ and $k=2$. We solve the problem of characterizing all connected graphs on $n$ vertices with $\rho_{o}(G)=n/\delta(G)$ which was presented in 2015 by Hamid and Saravanakumar. Also, by establishing a relation between the $k$-limited packing number and double domination number we improve two upper bounds given by Chellali and Haynes in 2005.