\begin{definition}[Definition:Coloring/Vertex Coloring]
A '''vertex $k$-coloring''' of a simple graph $G = \left({V, E}\right)$ is defined as an assignment of one element from a set $C$ of $k$ colors to each vertex in $V$.
That is, a '''vertex $k$-coloring''' of the graph $G = \left({V, E}\right)$ is a mapping $c: V \to \left\{{1, 2, \ldots k}\right\}$.
A graph with such a coloring is called a '''vertex-colored graph'''.
\end{definition}