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)| $\mathrm{.}Moreover,itturnsoutthat$\nu_{2}(G)\leq \frac{|V(G)|+2\nu_{3}(G)}{4}$.