  We prove that every connected cubic graph with $n$ vertices has a maximal
matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic
case of a conjecture of Baste, F\"urst, Henning, Mohr and Rautenbach (2019) on
regular graphs. More generally, we prove that every graph with $n$ vertices and
$m$ edges and maximum degree at most $3$ has a maximal matching of size at most
$\frac{4n-m}{6}+ \frac{1}{2}$. These bounds are attained by the graph
$K_{3,3}$, but asymptotically there may still be some room for improvement.
Moreover, the claimed maximal matchings can be found efficiently. As a
corollary, we have a $\left(\frac{25}{18} + O \left( \frac{1}{n}\right)\right)
$-approximation algorithm for minimum maximal matching in connected cubic
graphs.
