Edge-disjoint cycles with the same vertex set

Abstract

In 1975, Erdős 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 Turán-type results can be used to prove an upper bound of $n^{3/2+o\left(1\right)}$. However, this approach cannot give an upper bound better than $\Omega \left(n^{3/2}\right)$. We show that there is an absolute constant t and some constant $c=c\left(k\right)$ such that for each $k\ge 2$, every n-vertex graph with at least $cn{(\log n)}^t$ 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, Rödl and Szemerédi 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.