AbstractThe 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 Δ satisfies some basic expansion properties and contains a family of edge disjoint Hamilton cycles, then there also exists a covering of its edges by Hamilton cycles. This implies that for every α > 0 and every there exists a covering of all edges of G(n,p) by Hamilton cycles asymptotically almost surely, which is nearly optimal.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 183‐200, 2014