The order of the largest complete minor in a random graph

TL;DR

This paper precisely determines the size of the largest complete minor in a random graph G(n,p) for various ranges of p, extending previous results and answering an open question about the phase transition.

Contribution

It extends existing results to determine the contraction clique number ccl(G(n,p)) for p > C/n and resolves an open problem about its order at the phase transition point.

Findings

For p > C/n, ccl(G(n,p)) is tightly bounded.

At p = C/n with C > 1, ccl(G(n,p)) is asymptotically of order √n.

Answers an open question by Krivelevich and Sudakov.

Abstract

Let ccl(G) denote the order of the largest complete minor in a graph G (also called the contraction clique number) and let G(n,p) denote a random graph on n vertices with edge probability p. Bollobas, Catlin and Erdos asymptotically determined ccl(G (n,p)) when p is a constant. Luczak, Pittel and Wierman gave bounds on ccl(G(n,p)) when p is very close to 1/n, i.e. inside the phase transition. Extending the results of Bollobas, Catlin and Erdos, we determine ccl(G(n,p)) quite tightly, for p>C/n where C is a large constant. If p=C/n, for an arbitrary constant C>1, then we show that asymptotically almost surely ccl(G (n,p)) is of order square-root of n. This answers a question of Krivelevich and Sudakov.