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Graph Coloring Algorithms

A coloring of a graph is an assignment of labels to certain elements of a graph. More commonly, elements are either vertices (vertex coloring), edges (edge coloring), or both edges and vertices (total colorings). The most common form asks to color the vertices of a graph such that no two adjacent vertices share the same “color” (label). This is called a proper vertex coloring. For a graph G, the minimum number of colors such that G has a proper vertex coloring is denoted χ(G), the chromatic number of G. For example, for any path Pn on n vertices, χ(Pn) = 2. Observe that this bound also applies to any bipartite graph: we can assign color 1 to vertices of the first partition, and color 2 to vertices of the second, there will be no two adjacent vertices that share the same color since vertices of the same class are never adjacent to each other by definition of a bipartite graph. The lower bound on χ is obviously the clique number ω, but what about an upper bound? One would think about |V |, but other than the complete graph, this bound is never tight. With further thought, we can see that χ(G) ≤ ∆(G) + 1 (the proof is left to the reader). A result by Brooks states the following: