On k-(total) limited packing in graphs

A set B⊆ V(G) is called a k-total limited packing set in a graph G if |B∩ N(v)|≤ k for any vertex v∈ V(G). The k-total limited packing number Lk,t(G) is the maximum cardinality of a k-total limited packing set in G. Here, we give some results on the k-total limited packing number of graphs emphasizing trees, especially when $k=2$. We also study the $2$-(total) limited packing number of some product graphs. A k-limited packing partition (kLPP) of graph G is a partition of $V(G)$ into k-limited packing sets. The minimum cardinality of a kLPP is called the kLPP number of G and is denoted by χ× k(G), and we obtain some results for this parameter.