Let G be a bipartite graph and let A be the set of vertices of maximal degree. Show that there is a matching in G that covers A where A is not necessarily in one partite.
I am trying to apply a similar proof as that of Konig's Theorem but I am having trouble.
Add more edges (and maybe more vertices) until the bipartite graph is $\Delta(G)$-regular. What can you say about regular bipartite graphs?