Abstract
A k-limited packing partition (kLP partition) of a graph G is a partition of V(G) into k-limited packing sets. We consider the kLP partitions with minimum cardinality (with emphasis on k=2). The minimum cardinality is called kLP partition number of G and denoted by χ×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 χ×2 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 2TLP number and 2LP number with emphasis on trees.
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