More on limited packings in graphs

Abstract

A set B of vertices in a graph G is called a k-limited packing if for each vertex v of G, its closed neighbourhood has at most k vertices in B. The k-limited packing number of a graph G, denoted by \(L_k(G)\), is the largest number of vertices in a k-limited packing in G. The concept of the k-limited packing of a graph was introduced by Gallant et al., which is a generalization of the well-known packing of a graph. In this paper, we present some tight bounds for the k-limited packing number of a graph in terms of its order, diameter, girth, and maximum degree, respectively. As a result, we obtain a tight Nordhaus–Gaddum type result for the k-limited packing number. At last, we investigate the relationship among the open packing number, the packing number and 2-limited packing number of trees.

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Acknowledgements

The authors are very grateful to the reviewers for their useful suggestions and comments, which helped to improve the presentation of the paper.

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Correspondence to Xueliang Li.

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Bai, X., Chang, H. & Li, X. More on limited packings in graphs. J Comb Optim 40, 412–430 (2020). https://doi.org/10.1007/s10878-020-00606-z

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Keywords

  • k-limited packing
  • Opening packing
  • Nordhaus–Gaddum type result

Mathematics Subject Classification

  • 05C69
  • 05C70