  We show that any $2-$factor of a cubic graph can be extended to a maximum
$3-$edge-colorable subgraph. We also show that the sum of sizes of maximum $2-$
and $3-$edge-colorable subgraphs of a cubic graph is at least twice of its
number of vertices. Finally, for a cubic graph $G$, consider the pairs of
edge-disjoint matchings whose union consists of as many edges as possible. Let
$H$ be the largest matching among such pairs. Let $M$ be a maximum matching of
$G$. We show that 9/8 is a tight upper bound for $|M|/|H|$.
