On disjoint matchings in cubic graphs

For \(i=2,3\) and a cubic graph \(G\) let \(\nu_{i}(G)\) denote the maximum number of edges that can be covered by \(i\) matchings. We show that \(\nu_{2}(G)\geq {4/5}| V(G)| \) and \(\nu_{3}(G)\geq {7/6}| V(G)| \). Moreover, it turns out that \(\nu_{2}(G)\leq \frac{|V(G)|+2\nu_{3}(G)}{4}\).