Let be a simple bipartite graph, such that and and such that for every vertex in the graph, for some

Also, there is a closed path (the last vertex is the same as the first) of length that goes over every vertex in G at least once.

Prove that there is a perfect matching on G.

Pretty much at a loss on how I should do this. Any help is welcome.

I most likely need to use either Konig's theorem, or Hall's to solve this, but as I said I couldn't see where I could apply them.