How does the core sit inside the mantle?

Abstract: The $k$-core, defined as the largest subgraph of minimum degree $k$, of the random graph $G(n,p)$ has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [JCTB 67 (1996) 111--151] determined the threshold $d_k$ for the appearance of an extensive $k$-core. Here we derive a multi-type Galton-Watson branching process that describes precisely how the $k$-core is embedded into the random graph for any $k\geq3$ and any fixed average degree $d=np>d_k$. This generalises prior results on, e.g., the internal structure of the $k$-core.