Definition:Proper Coloring/Vertex Coloring
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Definition
Let $G = \struct {V, E}$ be a simple graph.
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:
- $\forall e = \set {u, v} \in E: \map c u \ne \map c v$