The order of the largest complete minor in a random graph

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. Bollobás, Catlin and Erdős [Bollobás B., P.A. Catlin, and P. Erdős, Hadwigerʹs conjecture is true for almost every graph, Europ. J. Combin. 1 (1980) 195–199] asymptotically determined ccl ( G n , p ) when p is a constant. Łuczak, Pittel and Wierman [Łuczak T., B. Pittel, and J.C. Wierman, The structure of a random graph at the point of the phase transition, Trans. Amer. Math. Soc. 341 (1994) 721–748] gave bounds on ccl ( G n , p ) when p is very close to 1/n, i.e. inside the phase transition. We show that for every ε > 0 there exists a constant C such that whenever C / n < p < I − ε then asymptotically almost surely ccl ( G n , p ) = ( 1 ± ε ) n / log b ( n p ) , where b : = 1 / ( 1 − p ) . If p = C / n for a constant C > 1 , then asymptotically almost surely ccl ( G n , p ) = Θ ( n ) . This extends the results in [Bollobás B., P.A. Catlin, and P. Erdős, Hadwigerʹs conjecture is true for almost every graph, Europ. J. Combin. 1 (1980) 195–199] and answers a question of Krivelevich and Sudakov [Krivelevich M., and B. Sudakov, Minors in expanding graphs, preprint 2006].