A simple solution to the k‐core problem

Abstract

ABSTRACT:We 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.

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“…Asymptotically c k ≈ √ k + k log k. The result applies without change to random bipartite graphs. This can be proved by making a small modi cation in the simpli ed proof of the basic result, which was given in [3]. Here the initial distribution of the degrees of the vertices is rst considered, for nite n the sum of the degrees is 2w, but asymptotically the distribution is Poisson.…”

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“…The proof of our theorem will be reminiscent of studies of the k-core, the pure literal rule, and other similar problems [14,11,6,7,9]. There, one repeatedly removes vertices (literals, etc.)…”

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“…Although the exact threshold for α k has been computed before in [6,16], we present a simpler analysis of the asymptotic formula. For a random graph on n nodes with k(1 − α)n edges, let us find the fraction f of nodes that have degree at most k − 1.…”