Edge-disjoint cycles with the same vertex set

In 1975, Erdos asked for the maximum number of edges that an n-vertex graph can have if it does not contain two edge-disjoint cycles on the same vertex set. It is known that Tur\'an-type results can be used to prove an upper bound of n^3/2+o (1). However, this approach cannot give an upper bound better than (n^3/2). We show that, for any k 2, every n-vertex graph with at least n polylog (n) edges contains k pairwise edge-disjoint cycles with the same vertex set, resolving this old problem in a strong form up to a polylogarithmic factor. The well-known construction of Pyber, R\"odl and Szemer\'edi of graphs without 4-regular subgraphs shows that there are n-vertex graphs with (n n) edges which do not contain two cycles with the same vertex set, so the polylogarithmic term in our result cannot be completely removed. Our proof combines a variety of techniques including sublinear expanders, absorption and a novel tool for regularisation, which is of independent interest. Among other applications, this tool can be used to regularise an expander while still preserving certain key expansion properties.