  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.
