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Limited packings: : Related vertex partitions and duality issues

Published: 01 July 2024 Publication History

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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Published In

cover image Applied Mathematics and Computation
Applied Mathematics and Computation  Volume 472, Issue C
Jul 2024
291 pages

Publisher

Elsevier Science Inc.

United States

Publication History

Published: 01 July 2024

Author Tags

  1. 05C15
  2. 05C69
  3. 05C76

Author Tags

  1. Limited packing
  2. 2-Limited packing partition number
  3. NP-hard
  4. Tuple domination
  5. Lexicographic product
  6. Nordhaus-Gaddum inequality

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