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

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