Limited packings: related vertex partitions and duality issues

TL;DR

This paper investigates the properties of limited packing partitions in graphs, providing exact values for trees, bounds for general graphs, and solving an open problem related to the dual of the problem, while also exploring computational complexity aspects.

Contribution

It introduces the concept of $k$LP partitions, determines exact values for trees, bounds for general graphs, and addresses an open problem in the dual problem, along with complexity results.

Findings

Exact $oldsymbol{ ext{chi}_{ imes 2}}$ for trees.

Bounds for $oldsymbol{ ext{chi}_{ imes 2}}$ in general graphs.

NP-hardness of computing total limited packing number.

Abstract

A $k$-limited packing partition ($k$LP partition) of a graph $G$ is a partition of $V(G)$ into $k$-limited packing sets. We consider the $k$LP partitions with minimum cardinality (with emphasis on $k=2$). The minimum cardinality is called $k$LP partition number of $G$ and denoted by $\chi_{\times k}(G)$. This problem is the dual problem of $k$-tuple domatic partitioning as well as a generalization of the well-studied $2$-distance coloring problem in graphs. We give the exact value of $\chi_{\times2}$ for trees and bound it for general graphs. A section of this paper is devoted to the dual of this problem, where we give a solution to an open problem posed in $1998$. We also revisit the total limited packing number in this paper and prove that the problem of computing this parameter is NP-hard even for some special families of graphs. We give some inequalities concerning this parameter and discuss the difference between $2$TLP number and $2$LP number with emphasis on trees.