Graph coloring

A graph coloring assigns labels called colours to elements of a graph, subject to specified constraints. Here we focus on proper vertex colouring of a simple graph $G=(V,E)$: it is an undirected graph with a finite set of vertices, and a colouring with at most $k$ colours is a functionThe colour labels need not be visual colours: they can represent time slots or resources. Unused labels are allowed. Properness concerns graph adjacency, so moving vertices in a drawing does not change whether the graph coloring is valid.^[ref]

Each colour class $c^{-1}(i)$ is an independent set: no edge has both endpoints in it. Conversely, a set partition of $V$ into independent sets gives a proper graph coloring by assigning one colour to each part. Thus graph coloring groups mutually compatible objects, and its chromatic number asks for the fewest such groups.

For an illustrative scheduling model, let each vertex represent an activity of equal duration and each edge represent a pair that cannot occur in the same time slot. A proper graph coloring is then a schedule respecting those pairwise conflicts. If the conflicts form a cycle graph $C_4$ with edges $12,23,34,41$, activities $1,3$ can share one slot and $2,4$ another. This model assumes interchangeable slots and includes only the pairwise conflicts encoded by the graph.^[ref]

The word "colouring" also occurs in edge coloring, which requires different colours on edges sharing an endpoint. The chromatic number discussed here always refers to proper vertex colouring.