Paired domination in graphs with minimum degree four

A set $S$ of vertices in a graph $G$ is a paired dominating set if every vertex of $G$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ admits a perfect matching. The minimum cardinality of a paired dominating set of $G$ is the paired domination number $\gpr(G)$ of $G$. We show that if $G$ is a graph of order~$n$ and $\delta(G) \ge 4$, then $\gpr(G) \le \frac{10}{17}n < 0.5883 n$.