  For some $\alpha$ with $0 < \alpha \le 1$, a subset $X$ of vertices in a
graph $G$ of order~$n$ is an $\alpha$-partial dominating set of $G$ if the set
$X$ dominates at least $\alpha \times n$ vertices in $G$. The $\alpha$-partial
domination number ${\rm pd}_{\alpha}(G)$ of $G$ is the minimum cardinality of
an $\alpha$-partial dominating set of $G$. In this paper partial domination of
graphs with minimum degree at least $3$ is studied. It is proved that if $G$ is
a graph of order~$n$ and with $\delta(G)\ge 3$, then ${\rm pd}_{\frac{7}{8}}(G)
\le \frac{1}{3}n$. If in addition $n\ge 60$, then ${\rm pd}_{\frac{9}{10}}(G)
\le \frac{1}{3}n$, and if $G$ is a connected cubic graph of order $n\ge 28$,
then ${\rm pd}_{\frac{13}{14}}(G) \le \frac{1}{3}n$. Along the way it is shown
that there are exactly four connected cubic graphs of order $14$ with
domination number $5$.
