# A bipartite graph coloring problem - maximizing number of colors

A problem I am looking at for my research inspired this problem.

Given a bipartite graph $G=(U,V,E)$, find $\max k$ such that there exists a $k$-coloring on V, $f:V \rightarrow \{1,2,\ldots,k\}$ which satisfies $\cup_{v\in \mathcal{N}(u)} f(v)=\{1,2,\ldots,k\}$, i.e. what is the maximum number of colors that can be assigned to vertices in $V$, such that all vertices in $U$ have all colors as neighbors?

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