Gallai Identities

Consider the following parameters of an undirected graph $G$ on $n$ vertices. - $\nu(G)$ is the size of a maximum matching of $G$. - $\tau(G)$ is the size of a minimum vertex cover of $G$. - $\alpha(G)$ is the size of a maximum independent set in $G$. - $\rho(G)$ is the size of a minimum edge cover of $G$. Prove the following : - $\nu(G) \leq \tau(G)$ - $\alpha(G) + \tau(G) = n$ - $\nu(G) + \rho(G) = n$ Prove the following when $G$ is a bipartite graph : - $\nu(G) = \tau(G)$ - $\rho(G) = \alpha(G)$ and hence conclude that the complement of a bipartite graph is [perfect](http://en.wikipedia.org/wiki/Perfect_graph).