Edge-disjoint cycles with the same vertex set
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910405472813056 |
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| author | Chakraborti, Debsoumya Janzer, Oliver Methuku, Abhishek Montgomery, Richard |
| author_facet | Chakraborti, Debsoumya Janzer, Oliver Methuku, Abhishek Montgomery, Richard |
| contents | 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 $Ω(n^{3/2})$. 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 $Ω(n\log \log n)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Edge-disjoint cycles with the same vertex set Chakraborti, Debsoumya Janzer, Oliver Methuku, Abhishek Montgomery, Richard Combinatorics 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 $Ω(n^{3/2})$. 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 $Ω(n\log \log n)$ 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. |
| title | Edge-disjoint cycles with the same vertex set |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.07190 |