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 b(k − 1)/2c edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size bn/2c. For k ≥ 3, Ak occurs in Gn,m,k with probability tending to 1 as n →∞, when 2m ≥ ckn for some suitable constant ck. ∗Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA. †School of Mathematical Sciences, University of North London, London N7 8DB, UK. Research supported by the STORM Research Group ‡Department of Computer Science, Birkbeck College, University of London §Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Research supported in part by NSF grant CCR-9530974