On the $k$-limited packing numbers in graphs

TL;DR

This paper establishes new bounds on the $k$-limited packing numbers in graphs, including sharp lower bounds and Nordhaus-Gaddum type bounds, and explores related concepts like packing and open packing numbers, with applications to domination numbers in trees.

Contribution

It provides the first sharp lower bounds on the lower $k$-limited packing number and a Nordhaus-Gaddum bound for the 2-limited packing number, extending existing graph theory results.

Findings

Sharp lower bounds on the lower $k$-limited packing number.

A Nordhaus-Gaddum type bound on the 2-limited packing number.

New upper bounds on domination and total domination numbers of trees.

Abstract

We give a sharp lower bound on the lower $k$-limited packing number of a general graph. Moreover, we establish a Nordhaus-Gaddum type bound on $2$-limited packing number of a graph. Also, we investigate the concepts of packing number ($1$-limited packing number) and open packing number in graphs with more details. In this way, by making use of the classic result of Meir and Moon (1975) and its total version (2005) we give new sharp upper bounds on domination number and total domination number of trees.