Introduction to Graph Coloring

A k-coloring of a graph G = (V,E) is a function c : V → C, where |C| = k. (Most often we use C = [k].) Vertices of the same color form a color class. A coloring is proper if adjacent vertices have different colors. A graph is k-colorable if there is a proper k-coloring. The chromatic number χ(G) of a graph G is the minimum k such that G is k-colorable. Let H and G be graphs. The disjoint union G+H of G and H is the graph whose vertices and edges are disjoint unions of vertices and edges of G and H , respectively. The join G ∨H of simple graphs G and H is obtain from G + H by adding all edges of the form (u, v), with u ∈ V (G), v ∈ V (H). The cartesian product G2H of G and H is the graph with vertex set V (G) × V (H) and edges of the form ((u, v), (u′, v′)), where either u = u′ and (v, v′) ∈ E(H), v = v′ and (u, u′) ∈ E(G).