Upper-bounding the k-colorability threshold by counting covers

Abstract

Let G(n, m) be the random graph on n vertices with m edges. Let d=2m/n be its average degree. We prove that G(n, m) fails to be k-colorable with high probability if d>2k ln k-ln k-1+ok(1). This matches a conjecture put forward on the basis of sophisticated but non-rigorous statistical physics ideas (Krzakala, Pagnani, Weigt: Phys. Rev. E 70 (2004)). The proof is based on applying the first moment method to the number of "covers", a physics-inspired concept. By comparison, a standard first moment over the number of k-colorings shows that G(n, m) is not k-colorable with high probability if d > 2k ln k - ln k.