HITTING TIME OF EDGE DISJOINT HAMILTON CYCLES IN RANDOM SUBGRAPH PROCESSES ON DENSE BASE GRAPHS

Consider the random subgraph process on a base graph G on n vertices: a sequence {G(t)}(t=0)(vertical bar E(G)vertical bar) of random subgraphs of G obtained by choosing an ordering of the edges of G uniformly at random, and by sequentially adding edges to G(0), the empty graph on the vertex set of G, according to the chosen ordering. We show that if G has one of the following properties: 1. there is a positive constant epsilon > 0 such that delta(G) >= (1/2 + epsilon) n; 2. there are some constants alpha, beta > 0 such that every two disjoint subsets U, W of size at least an have at least beta vertical bar U vertical bar vertical bar W vertical bar edges between them, and the minimum degree of G is at least (2 alpha + beta) center dot n; or 3. G is an (n, d, lambda)-graph, with d >= C.n,log log n/log n and lambda <= c.d(2)/n for some absolute constants c, C > 0; then for a positive integer constant k with high probability the hitting time of the property of containing k edge disjoint Hamilton cycles is equal to the hitting time of having minimum degree at least 2k. These results extend prior results by Johansson and by Frieze and Krivelevich and answer a question posed by Frieze.