  The \emph{minimum leaf number} $\hbox{ml} (G)$ of a connected graph $G$ is
defined as the minimum number of leaves of the spanning trees of $G$. We
present new results concerning the minimum leaf number of cubic graphs: we show
that if $G$ is a connected cubic graph of order $n$, then $\mathrm{ml}(G) \leq
\frac{n}6 + \frac13$, improving on the best known result in [Inf. Process.
Lett. 105 (2008) 164-169] and proving the conjecture in [Electron. J. Graph
Theory and Applications 5 (2017) 207-211]. We further prove that if $G$ is also
2-connected, then $\mathrm{ml}(G) \leq \frac{n}{6.53}$, improving on the best
known bound in [Math. Program., Ser. A 144 (2014) 227-245]. We also present new
conjectures concerning the minimum leaf number of several types of cubic graphs
and examples showing that the bounds of the conjectures are best possible.
