Minors in random regular graphs

We show that there is a constant c so that for fixed r ≥ 3 a.a.s. an r ‐regular graph on n vertices contains a complete graph on $ c\sqrt{n} $ vertices as a minor. This confirms a conjecture of Markström (Ars Combinatoria 70 (2004) 289–295). Since any minor of an r ‐regular graph on n vertices has at most r n /2 edges, our bound is clearly best possible up to the value of the constant c . As a corollary, we also obtain the likely order of magnitude of the largest complete minor in a random graph G n , p during the phase transition (i.e., when p n → 1). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009