  The problem of packing Hamilton cycles in random and pseudorandom graphs has
been studied extensively. In this paper, we look at the dual question of
covering all edges of a graph by Hamilton cycles and prove that if a graph with
maximum degree $\Delta$ satisfies some basic expansion properties and contains
a family of $(1-o(1))\Delta/2$ edge disjoint Hamilton cycles, then there also
exists a covering of its edges by $(1+o(1))\Delta/2$ Hamilton cycles. This
implies that for every $\alpha >0$ and every $p \geq n^{\alpha-1}$ there exists
a covering of all edges of $G(n,p)$ by $(1+o(1))np/2$ Hamilton cycles
asymptotically almost surely, which is nearly optimal.
