Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least k

Let Gn,m,k denote the space of simple graphs with n vertices, m edges, and minimum degree at least k, each graph G being equiprobable. Let G have property Ak, if G contains [(k - 1)/2] edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size [n/2]. We prove that, for k greater than or equal 3, there is a constant Ck such that if 2m greater than or equal Ckn then Ak occurs in Gn,m,k with probability tending to 1 as n -> infinity . © 2000 John Wiley and Sons, Inc. J Graph Theory.