How does the core sit inside the mantle?

TL;DR

This paper develops a detailed probabilistic model describing how the $k$-core is embedded within random graphs for degrees above the threshold, extending previous structural analyses.

Contribution

It introduces a multi-type Galton-Watson process to precisely characterize the $k$-core's internal structure in random graphs for all $k \\geq 3$ and fixed average degrees.

Findings

Derived a multi-type Galton-Watson process for $k$-core embedding

Generalized prior results on $k$-core structure

Provides a detailed probabilistic description of the $k$-core within random graphs

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.