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We study the k-core of a random (multi)graph on nvertices with a given degree sequence. We let n→∞. Then, under some regularity conditions on thedegree sequences, we give conditions on the asymptotic shape of thedegree sequence that imply that with high probability thek-core is empty and other conditions that imply that withhigh probability the k-core is non-empty and the sizes ofits vertex and edge sets satisfy a law of large numbers; undersuitable assumptions these are the only two possibilities. Inparticular, we recover the result by Pittel, Spencer, and Wormald(J Combinator Theory 67 (1996), 111151) on the existence and sizeof a k-core in G(n,p) andG(n,m), see also Molloy (Random Struct Algor27 (2005), 124135) and Cooper (Random Struct Algor 25 (2004),353375). Our method is based on the properties of empiricaldistributions of independent random variables and leads to simpleproofs. © 2006 Wiley Periodicals, Inc. Random Struct. Alg.,,2007