# Maximum "k-to-k" matching in bipartite graph

I was "inspired" to extend the concept of Maximum "$2$-to-$1$" matching in a bipartite graph 2-to-1 matching by extending it to "k-to-k" matching for some natural number k.

My strategy is as follows: Begin with $$G = (L \dot \lor R, E)$$ and find typical 1-1 matching here. Then, construct $$G' = (L \dot \lor R, E')$$ only with the edges that were not included in the matching from the previous step, and find the maximum matching for $$G'$$. Similarly, $$G''$$ will have only edges which were in neither of the maximum matchings from before. Repeat this procedure until $$G^{(k-1)}$$.

I cannot find a counter example to disprove this logic, but I am unable to prove it either.

My question is, is there something fundamental that I am missing here? Does this algorithm work?

This has a "$$2$$-to-$$2$$ matching". But if you start with the matching consisting of the vertical edges, the remaining graph does not have a perfect matching.
One the other hand, it is true that for some choices of perfect matchings, your method will return a $$k$$-to-$$k$$ matching, if there is one. This follows from (what I know as) Kőnig's theorem, which says that a $$k$$-regular bipartite graph can be decomposed into the union of $$k$$ perfect matchings.
By the way, the usual term for what you call a $$k$$-to-$$k$$ matching is a "$$k$$-factor".