Scheduling problem on bipartite graph
So here, it looks as though $u \in U$ is a vertex representing the team and $v \in V$ is an edge between the vertices representing the matches possible.

If we can generate a Bipartite Graph, the edges ${ v_i, ...v_j}$  going across are all non-conflicting matches. Those edges for this matching ( $M_1$) can be held on that day. Next, we remove ${ v_i, ...v_j}$ and check to see if we can get more Bipartite Graphs. The edges going across are similarly the non-conflicting matches. 

Not sure if this is what you asked for.