Edgedisjoint 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 edgedisjoint cycles on the same vertex set. It is known that Turántype 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 $\Omega(n^{3/2})$. We show that, for any $k\geq 2$, every $n$vertex graph with at least $n \cdot \mathrm{polylog}(n)$ edges contains $k$ pairwise edgedisjoint cycles with the same vertex set, resolving this old problem in a strong form up to a polylogarithmic factor. The wellknown 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.07190
 arXiv:
 arXiv:2404.07190
 Bibcode:
 2024arXiv240407190C
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 34 pages, 2 figures