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 Δ satisfies some basic expansion properties and contains
a family of (1−o(1))Δ/2 edge disjoint Hamilton cycles, then there also
exists a covering of its edges by (1+o(1))Δ/2 Hamilton cycles. This
implies that for every α>0 and every p≥nα−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.Comment: 19 pages. arXiv admin note: some text overlap with arXiv:some
math/061275