Consider a bipartite graph $(G, U, V)$. 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?
I am afraid that the sheduling problem is equivalent to the coloring problem (which is NP-hard). Indeed, construct new graph $G’$ over the set of vertices $U$. We join vertices $u_i$ and $u_j$ in $G’$ by an edge iff they have a common neighbor in the graph $G$. Then we have to find a coloring of the graph $G’$ realizing its chromatic number. Vise versa, the coloring problem for a graph $G’=(U,E)$ can be reduced to the sheduling problem for a bipartite graph $(G, E, U)$, where a vertex $u\in U$ is joined with $e\in E$ iff $u$ is incident on an edge $e$ in the graph $G’$.