I am working on a problem that involves finding the minimum number of colors to color the edges of a bipartite graph with N vertices on each side subject to a few conditions. I will call each vertex on the left $L_i$ and each on the right $R_i$.
Each color has a subset of indices for the left hand side and the right hand side. Say our first color is $C_1 = ({1, 2, 5}), ({3, 6, 7})$. Then, every index on the left subset would have to connect to every index on the right subset. $L_1$ would have to connect to $R_3, R_6, R_7$ for example, as would $L_2$ and $L_5$.
What makes this hard for me is that every vertex on the left must connect to every vertex on the right once, apart from the missing edge between $L_i, R_j$ where i = j. This means that the two subsets that represent a color must have an empty intersection, or else $L_i$ and $R_i$ would have an edge between them.
Intuitively, I think the answer is N since I can't think an example where N-1 is possible, but I am not too sure about how to start formalizing it. Any hints on where to head next?
