Scheduling problem on bipartite graph Consider a bipartite graph $G=(U, V, E)$. Each $v \in V$ represents a soccer team, and each $u \in U$ represents a mini-tournament needs to be scheduled. 

If $u_i$ and $u_j$ share no common neighbor, these two tournament can be scheduled on the same day. Similarly, one can schedule multiple tournaments in one day if there is no such conflict. 

Is there a way to compute the minimum number of days required to complete all the tournaments and the corresponding scheduling?
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.