A simple solution to the k‐core problem

Abstract

AbstractWe study the k‐core of a random (multi)graph on n vertices with a given degree sequence. We let n →∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k‐core is empty and other conditions that imply that with high probability the k‐core is non‐empty and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer, and Wormald (J Combinator Theory 67 (1996), 111–151) on the existence and size of a k‐core in G(n,p) and G(n,m), see also Molloy (Random Struct Algor 27 (2005), 124–135) and Cooper (Random Struct Algor 25 (2004), 353–375). Our method is based on the properties of empirical distributions of independent random variables and leads to simple proofs. © 2006 Wiley Periodicals, Inc. Random Struct. Alg.,, 2007