Is this problem just an application of traveling salesman? If not is it some other already "solved" problem?
This problem comes down to a transportation problem (actually the Wikipedia article is not very good), although it is not exactly the problem you are stating. As David Richerby stated in the comments, your problem is formulated contradictory, therefore I would suggest a reformulation, where you fix the number of vertices that each path visits. But while reading yourself into the capacitive transport problem, you will find out yourself what changes fit best for your task.
Solving these problems can indeed, as stated by Raphael, be solved via min-cost max-flow algorithms. 
When you decided on a new version I am happy to extend my answer to fit your needs.