Edge-disjoint Hamilton cycles in graphs

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every α>0, every sufficiently large graph on n vertices with minimum degree at least (1/2+α)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree δ must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every α>0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least (1/2+α)n can be almost decomposed into edge-disjoint Hamilton cycles.