Definition:Proper Coloring

A k-coloring of a graph $$G = (V,E)$$ is defined as an assignment of one color from a set of k colors to each vertex in $$V$$ such that no two neighboring vertices share a common color.

That is, a k-coloring of the graph $$G = (V,E)$$ is a functional mapping $$c : V \to \{1, 2, \ldots k \}$$ such that, $$\forall e = \{u, v\} \in E, c(u) \neq c(v)$$.