I have a graph $G(V,E)$ where each vertex has degree at most $k$. How can I prove that I need at most $2k + 1$ disjoint sets of vertices in which there is no connection between two vertices from the same set to create the set of vertices $V$.
closed as off-topic by Namaste, Morgan Rodgers, Aqua, Misha Lavrov, Rolf Hoyer Nov 12 '17 at 14:20
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It is well-known and easy to prove that if each vertex of a graph has degree at most $k$ then its vertices can be colored into $k+1$ colors in such a way that there are no adjacent monochromatic vertices. This yield a required partition even into $k+1$ sets of vertices.