The k-limited packing number, Lk(G), of a graph G, introduced by
Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set X
of vertices of G such that every vertex of G has at most k elements of
X in its closed neighbourhood. The main aim in this paper is to prove the
best-possible result that if G is a cubic graph, then L2(G)≥∣V(G)∣/3, improving the previous lower bound given by Gallant, \emph{et al.}
In addition, we construct an infinite family of graphs to show that lower
bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a
constant factor, when k is fixed and Δ(G) tends to infinity. For
Δ(G) tending to infinity and k tending to infinity sufficiently
quickly, we give an asymptotically best-possible lower bound for Lk(G),
improving previous bounds