On disjoint matchings in cubic graphs: Maximum 2-edge-colorable and maximum 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 GG, consider the pairs of edge-disjoint matchings whose union consists of as many edges as possible. Let HH be the largest matching among such pairs. Let MM be a maximum matching of GG. We show that 9/89/8 is a tight upper bound for |M|/|H||M|/|H|.