Chasing the K-Colorability Threshold

In this paper we establish a substantially improved lower bound on the k-color ability threshold of the random graph G(n, m) with n vertices and m edges. The new lower bound is ≈ 1.39 less than the 2k ln (k)-ln (k) first-moment upper bound (and approximately 0.39 less than the 2k ln (k) - ln(k) - 1 physics conjecture). By comparison, the best previous bounds left a gap of about 2+ln(k), unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004]. Furthermore, we prove that, in a precise sense, our lower bound marks the so-called condensation phase transition predicted on the basis of physics arguments [Krzkala et al.: PNAS 2007]. Our proof technique is a novel approach to the second moment method, inspired by physics conjectures on the geometry of the set of k-colorings of the random graph.