Sandwiching a densest subgraph by consecutive cores

Abstract

In this paper, we show that in the random graph G(n, c/n), with high probability, there exists an integer k such that a subgraph of G(n, c/n), whose vertex set differs from a densest subgraph of G(n, c/n) by O(log 2 n) vertices, is sandwiched by the k and the ( k + 1)-core, for almost all sufficiently large c. We determine the value of k. We also prove that (a), the threshold of the k-core being balanced coincides with the threshold that the average degree of the k-core is at most 2(k − 1), for all sufficiently large k; (b) with high probability, there is a subgraph of G(n, c/n) whose density is significantly denser than any of its nonempty cores, for almost all sufficiently large c > 0.