On disjoint matchings in cubic graphs

For i = 2 , 3 and a cubic graph G let ν i ( G ) denote the maximum number of edges that can be covered by i matchings. We show that ν 2 ( G ) ≥ 4 5 | V ( G ) | and ν 3 ( G ) ≥ 7 6 | V ( G ) | . Moreover, it turns out that ν 2 ( G ) ≤ | V ( G ) | + 2 ν 3 ( G ) 4 .