On disjoint matchings in cubic graphs: maximum 2- and 3-edge-colorable subgraphs

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 $\mathrm{\mid }M\mathrm{\mid }\mathrm{/}\mathrm{\mid }H\mathrm{\mid }$.