Title: Edge-disjoint cycles with the same vertex set
Abstract: 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 Turan-type results can be used to prove an upper bound n^{3/2+o(1)}. However, this approach cannot give an upper bound better than n^{3/2}. We show that, for any k, 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, Rodl and Szemeredi of graphs without 4-regular subgraphs shows that there are n-vertex graphs with \Omega(n log log 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.
Joint work with Debsoumya Chakraborti, Abhishek Methuku and Richard Montgomery.
