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

Let G_n,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 A_k, 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 ≥ 3, there is a constant C_k such that if 2m ≥ C_kn then A_k occurs in G_n,m,k with probability tending to 1 as n → ∞. © 2000 John Wiley & Sons, Inc. J. Graph Theory 34: 42–59, 2000