On k-(total) limited packing in graphs

A set $B\subseteq V(G)$ is called a $k$-total limited packing set in a graph $G$ if $\mathrm{\mid }B\cap N(v)\mathrm{\mid }\le k$ for any vertex $v\in V(G)$. The $k$-total limited packing number $L_{k,t}(G)$ 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 ($k$LPP) of graph $G$ is a partition of $V(G)$ into $k$-limited packing sets. The minimum cardinality of a $k$LPP is called the $k$LPP number of $G$ and is denoted by ${\chi }_{\times k}(G)$, and we obtain some results for this parameter.