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

Phase transitions in graphs on orientable surfaces

“…More generally, given k ≥ 2, the k-core of a graph G is the largest subgraph of G of minimum degree at least k. Like the core, the k-core can be constructed by a peeling process that recursively removes vertices of degree less than k. The order and size of the k-core of G(n, m) has been determined in a seminal paper by Pittel, Spencer, and Wormald [58]. Following Pittel, Spencer, and Wormald, the k-core has been extensively studied [23,24,41,46,49,60]. The most striking results in this area are the astonishing theorem by Luczak [49] that the k-core for k ≥ 3 jumps to linear order at the very moment it becomes non-empty, the central limit theorem by Janson and Luczak [41], and the local limit theorem by Coja-Oghlan, Cooley, Kang, and Skubch [23] that described-in addition to the order and size-several other parameters of the k-core of G(n, m).…”

“…Following Pittel, Spencer, and Wormald, the k-core has been extensively studied [23,24,41,46,49,60]. The most striking results in this area are the astonishing theorem by Luczak [49] that the k-core for k ≥ 3 jumps to linear order at the very moment it becomes non-empty, the central limit theorem by Janson and Luczak [41], and the local limit theorem by Coja-Oghlan, Cooley, Kang, and Skubch [23] that described-in addition to the order and size-several other parameters of the k-core of G(n, m). In [24], the same authors used a 5-type branching process in order to determine the local structure of the k-core.…”

“…In [24], the same authors used a 5-type branching process in order to determine the local structure of the k-core. In terms of global structure, [23] provides a randomised algorithm that constructs a random graph with given order and size of the k-core.…”