Let $G=(V,E)$ be a graph whose vertices can be colored with $m$-colors, such that each color appears at least twice.
Let $\lambda(G)$ denote the minimum number of colors needed for a vertex-coloring of $G$
Use 1. to prove that such a coloring can be achieved with $\lambda(G)$ colors
Let $\alpha$ denote a coloring with property 1. and $\beta$ a coloring with $\lambda(G)$ colors. $\beta$ has a color class with only one vertex if $\beta$ doesn't have property 1.
I don't know how that helps to prove that such a coloring can be achieved with $\lambda(G)$ colors
Edit: A coloring of the graph $G$ is a proper coloring. $u,v$ have different colors if $(u,v)\in E$
Would appreciate any help