Definition:Proper Coloring

Proper Vertex Coloring
A proper (vertex) $$k$$-coloring of a graph $$G = \left({V, E}\right)$$ 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 $$G = \left({V, E}\right)$$ is a mapping $$c: V \to \left\{{1, 2, \ldots k}\right\}$$ such that:
 * $$\forall e = \left\{{u, v}\right\} \in E: c \left({u}\right) \ne \left({v}\right)$$.

Proper Edge Coloring
A proper (edge) $$k$$-coloring of a graph $$G = \left({V, E}\right)$$ is defined as an assignment of one color from a set of $$k$$ colors to each edge in $$E$$ such that no two adjacent edges share a common color.

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