Core forging and local limit theorems for the k-core of random graphs

Abstract: We establish a multivariate local limit theorem for the order and size as well as several other parameters of the k-core of the Erdos-Renyi graph. The proof is based on a novel approach to the k-core problem that replaces the meticulous analysis of the peeling process by a generative model of graphs with a core of a given order and size. The generative model, which is inspired by the Warning Propagation message passing algorithm, facilitates the direct study of properties of the core and its connections with the mantle and should therefore be of interest in its own right.