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 Gn,p denote a random graph on n vertices with edge probability p. Bollob´as, Catlin and Erd˝os [5] asymptotically determined ccl(Gn,p) when p is a constant. Luczak, Pittel and Wierman [10] gave bounds on ccl(Gn,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 < 1 − ε then asymptotically almost surely ccl(Gn,p)= (1 ± ε)n/ plogb(np), where b := 1/(1 − p). If p = C/n for a constant C > 1, then asymptotically almost surely ccl(Gn,p)= Θ( √ n). This extends the results in [5] and answers a question of Krivelevich and Sudakov [9].

• Type: Journal
• Year: 2007
• Where: ENDM
• Authors: Nikolaos Fountoulakis, Daniela Kühn, Deryk Osthus
• Keywords: Contraction Clique Number, Edge Probability, ENDM 2007, Random Graph