Sudden Emergence of a Giantk-Core in a Random Graph

Thek-core of a graph is the largest subgraph with minimum degree at leastk. For the Erdo?s?R?nyi random graphG(n,?m) onnvertives, withmedges, it is known that a giant 2-core grows simultaneously with a giant component, that is, whenmis close ton/2. We show that fork?3, with high probability, a giantk-core appears suddenly whenmreachesckn/2; hereck=min?>0?/?k(?) and?k(?)=P{Poisson(?)?k?1}. In particular,c3?3.35. We also demonstrate that, unlike the 2-core, when ak-core appears for the first time it is very likely to be giant, of size ?pk(?k)n. Here?kis the minimum point of?/?k(?) andpk(?k)=P{Poisson(?k)?k}. Fork=3, for instance, the newborn 3-core contains about 0.27nvertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one.

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