How does the core sit inside the mantle?

The k‐core, defined as the maximal subgraph of minimum degree at least k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold dk for the appearance of an extensive k‐core. The aim of the present paper is to describe how the k‐core is “embedded” into the random graph in the following sense. Let k≥3 and fix d=np>dk. Colour each vertex that belongs to the k‐core of G(n,p) in black and all remaining vertices in white. Here we derive a multi‐type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k‐core. In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2‐core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k‐core. Based on this the study of the k‐core reduces to the analysis of Warning Propagation on a suitable Galton‐Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 459–482, 2017