Definition:Proper Coloring

Definition
Let $G = \struct {V, E}$ be a simple graph.

Proper Vertex Coloring
A proper (vertex) $k$-coloring of $G$ is defined as a vertex coloring from a set of $k$ colors such that no two adjacent vertices share a common color.

That is, a proper $k$-coloring of $G$ is a mapping $c: V \to \set {1, 2, \ldots k}$ such that:
 * $\forall e = \set {u, v} \in E: \map c u \ne \map c v$

Proper Edge Coloring
A proper (edge) $k$-coloring of $G$ is defined as an edge coloring from a set of $k$ colors such that no two adjacent edges share a common color.

That is, a proper $k$-coloring of $G$ is a mapping $c: E \to \set {1, 2, \ldots k}$ such that:
 * $\forall v \in V: \forall e = \set {u_k, v} \in E: \map c {\set {u_i, v} } \ne \map c {\set {u_j, v} }$