Let $G$ be a bipartite graph in which the greatest degree of a vertex is 2019. Let $m$ be the least natural number for which we can color the edges of $G$ in $m$ colors so that each two edges with a common vertex from $G$ are in different colors. Show that $m$ doesn’t depend on $G$ and find its value.
Pluto1708, we are coloring the edges, not the vertices. The edges are colored in such way that no two of them, that are in the same color, share a common vertex from G.
This is a Konig result (theorem) and is present in every decent graph theory book.
Every bipartite graph with max degree $\Delta$ has an edge coloring with $\Delta$ colors. (Less than $\Delta$ colors obviously cannot work).
See R. Diestel, Graph theory, 5.3. The proof is instructive, the idea used in many other situations.