Speaker: Abhishek Methuku (ETH Zürich)
Title: 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. This problem has since been reiterated by several authors in various different contexts, including Bollobás in 1978 in his popular book 'Extremal graph theory', Pyber, Rödl, and Szemerédi in 1995, and Chen, Erdős, and Staton in 1996.
We asymptotically resolve this problem in a strong form, by showing that, for any given k the maximum number of edges in an n-vertex graph not containing k edge-disjoint cycles on the same vertex set is n^{1+o(1)}.
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, Oliver Janzer and Richard Montgomery.
