On disjoint matchings in cubic graphs

For i=2,3i=2,3 and a cubic graph GG let νi(G)νi(G) denote the maximum number of edges that can be covered by ii matchings. We show that ν2(G)≥45|V(G)| and ν3(G)≥76|V(G)|. Moreover, it turns out that ν2(G)≤|V(G)|+2ν3(G)4.