The probabilistic approach to limited packings in graphs

Abstract

We consider (closed neighbourhood) packings and their generalization in graphs. A vertex set X in a graph G is a k-limited packing if for every vertex v∈V(G), |N[v]∩X|≤k, where N[v] is the closed neighbourhood of v. The k-limited packing numberLk(G) of a graph G is the largest size of a k-limited packing in G. Limited packing problems can be considered as secure facility location problems in networks.In this paper, we develop a new application of the probabilistic method to limited packings in graphs, resulting in lower bounds for the k-limited packing number and a randomized algorithm to find k-limited packings satisfying the bounds. In particular, we prove that for any graph G of order n with maximum vertex degree Δ, Lk(G)≥kn(k+1)(Δk)(Δ+1)k. Also, some other upper and lower bounds for Lk(G) are given.