Sandwiching a densest subgraph by consecutive cores

Abstract

In this paper, we show that in the random graph Gn,c/n, with high probability, there exists an integer ki¾? such that a subgraph of Gn,c/n, whose vertex set differs from a densest subgraph of Gn,c/n by Olog2n vertices, is sandwiched by the ki¾? and the ki¾?+1-core, for almost all sufficiently large c. We determine the value of ki¾?. 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 2k-1, for all sufficiently large k; b with high probability, there is a subgraph of Gn,c/n whose density is significantly denser than any of its nonempty cores, for almost all sufficiently large c > 0. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 341-360, 2015