Coloring 3-Colorable Graphs

Graph coloring in general is an extremely easy-to-understand yet powerful tool. It has wide-ranging applications from register allocation to image segmentation. For such a simple problem, however, the question is surprisingly intractable. In this section I will introduce the problem formally, as well as present some general background on graph coloring. There are several ways to color a graph, in particular, one can color the vertices, faces, or the edges. These problems turn out to be equivalent for example, coloring the faces of a graph is the same as coloring the vertices of the dual, while coloring the edges is the same as coloring the vertices of the line graph. Today we consider the problem of vertex coloring. For the sake of formality, here are a few definitions. Definition A coloring for a graph G = (V,E) is a set of colors C along with a function f mapping the vertex set V into C. Definition A coloring is legal if ∀i, j ∈ V, (i, j) ∈ E ⇒ f(i) 6= f(j). In other words, one cannot color adjacent vertices the same color. Definition A graph is n-colorable if there exists a legal coloring on n colors. Definition The chromatic number of a graph is the minimum cardinality over all sets of colors that admit a legal coloring. Already, we have the following theorem. Theorem 1.1. Determining the chromatic number of a graph is NP-complete. It turns out the situation is even more dire. Theorem 1.2. Let n be the chromatic number of a graph. Then approximating n to within n1− for > 0 is NP-hard. With this in mind, we turn to a slightly easier question: assuming we know that a graph is 3-colorable, what’s the best we can do?