I have a simple bipartite graph with bipartition (W, C) such that $|W| < |C|$, each vertex $c \in C$ has 2 edges (so to two different vertices w each time) and each vertex $w \in W$ has 4 edges. I would like to prove the existence of groups that are made of 2 vertices $w$ each time, where both vertices have an edge to the same vertex $c$. Here is an example of such Graph.
This problem is just an example with values of a bigger problem where I need to look for the existence of groups that give me the smallest possible number of groups, and I have different values instead of 2 and 4 for the number of edges of vertices in C or W. So if you have any idea/hint/it makes you think of some problem: it is highly welcomed!