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 $n^{3/2+o(1)}$. 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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2024
- DOI:
- 10.48550/arXiv.2404.07190
- arXiv:
- arXiv:2404.07190
- Bibcode:
- 2024arXiv240407190C
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 34 pages, 2 figures