Cs880: Approximations Algorithms 21.1 Graph Coloring: Overview 21.2 Sdp Formulation

Let G = (V,E) be a graph. A k-coloring for G is a function f : V → [k] such that f(u) 6= f(v) for all (u, v) ∈ E. In other words, a k-coloring is an assignment of vertices to k colors such that no edge is monochromatic. We say that a graph G is k-colorable if there exists a k-coloring for G. The chromatic number of G is the least k such that G is k-colorable. Given a k-colorable graph G, finding a k-coloring for G is solvable in polynomial time for k = 2, but NP-hard for k ≥ 3.