On k-(total) limited packing in graphs

TL;DR

This paper investigates the properties of k-total limited packing sets in graphs, focusing on trees and product graphs, and introduces the concept of k-limited packing partitions with new bounds and results.

Contribution

It provides new bounds and results for the k-total limited packing number and k-limited packing partitions in graphs, especially for trees and certain product graphs.

Findings

Determined bounds for the k-total limited packing number in trees.

Analyzed the 2-total limited packing number in product graphs.

Established results for the k-limited packing partition number.

Abstract

A set $B\subseteq V(G)$ is called a $k$-total limited packing set in a graph $G$ if $|B\cap N(v)|\leq k$ for any vertex $v\in V(G)$. The $k$-total limited packing number $L_{k,t}(G)$ is the maximum cardinality of a $k$-total limited packing set in $G$. Here, we give some results on the $k$-total limited packing number of graphs emphasizing trees, especially when $k=2$. We also study the $2$-(total) limited packing number of some product graphs. A $k$-limited packing partition ($k$LPP) of graph $G$ is a partition of $V(G)$ into $k$-limited packing sets. The minimum cardinality of a $k$LPP is called the $k$LPP number of $G$ and is denoted by $\chi_{\times k}(G)$, and we obtain some results for this parameter.