More on limited packings in graphs

TL;DR

This paper studies the $k$-limited packing number in graphs, providing tight bounds based on various graph parameters and exploring its relationships with other packing concepts, including a Nordhaus-Gaddum-type result.

Contribution

It offers new tight bounds for the $k$-limited packing number and establishes its relationships with other packings, extending the understanding of this graph parameter.

Findings

Derived tight bounds for $L_k(G)$ based on order, diameter, girth, and maximum degree.

Established a Nordhaus-Gaddum-type result for the $k$-limited packing number.

Explored relationships among open packing, packing, and 2-limited packing in trees.

Abstract

A set $B$ of vertices in a graph $G$ is called a \emph{$k$-limited packing} if for each vertex $v$ of $G$, its closed neighbourhood has at most $k$ vertices in $B$. The \emph{$k$-limited packing number} of a graph $G$, denoted by $L_k(G)$, is the largest number of vertices in a $k$-limited packing in $G$. The concept of the $k$-limited packing of a graph was introduced by Gallant et al., which is a generalization of the well-known packing of a graph. In this paper, we present some tight bounds for the $k$-limited packing number of a graph in terms of its order, diameter, girth, and maximum degree, respectively. As a result, we obtain the tight Nordhaus-Gaddum-type result of this parameter for general $k$. At last, we investigate the relationship among the open packing number, the packing number and $2$-limited packing number of trees.