A note on the k-tuple domination number of graphs

In a graph $G$, a vertex dominates itself and its neighbours. A set $D\subseteq V(G)$ is said to be a $k$-tuple dominating set of $G$ if $D$ dominates every vertex of $G$ at least $k$ times. The minimum cardinality among all $k$-tuple dominating sets is the $k$-tuple domination number of $G$. In this paper, we provide new bounds on this parameter. Some of these bounds generalize other ones that have been given for the case $k=2$. In addition, we improve two well-known lower bounds on the $k$-tuple domination number.