  In a recent paper, we showed that every sufficiently large regular digraph G
on n vertices whose degree is linear in n and which is a robust outexpander has
a decomposition into edge-disjoint Hamilton cycles. The main consequence of
this theorem is that every regular tournament on n vertices can be decomposed
into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large.
This verified a conjecture of Kelly from 1968. In this paper, we derive a
number of further consequences of our result on robust outexpanders, the main
ones are the following: (i) an undirected analogue of our result on robust
outexpanders; (ii) best possible bounds on the size of an optimal packing of
edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range
of values for d. (iii) a similar result for digraphs of given minimum
semidegree; (iv) an approximate version of a conjecture of Nash-Williams on
Hamilton decompositions of dense regular graphs; (v) the observation that dense
quasi-random graphs are robust outexpanders; (vi) a verification of the `very
dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint
Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the
size of an optimal packing of edge-disjoint Hamilton cycles in a random
tournament.
