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 adjacent vertices share a common color.

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