A proof of a conjecture of Erdős and Gyárfás on monochromatic path covers
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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_ | 1866918155252662272 |
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| author | Pokrovskiy, Alexey Versteegen, Leo Williams, Ella |
| author_facet | Pokrovskiy, Alexey Versteegen, Leo Williams, Ella |
| contents | In 1995, Erdős and Gyárfás proved that in every $2$-edge-coloured complete graph on $n$ vertices, there exists a collection of $2\sqrt{n}$ monochromatic paths, all of the same colour, which cover the entire vertex set. They conjectured that it is possible to replace $2\sqrt{n}$ by $\sqrt{n}$. We prove this to be true for all sufficiently large $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A proof of a conjecture of Erdős and Gyárfás on monochromatic path covers Pokrovskiy, Alexey Versteegen, Leo Williams, Ella Combinatorics In 1995, Erdős and Gyárfás proved that in every $2$-edge-coloured complete graph on $n$ vertices, there exists a collection of $2\sqrt{n}$ monochromatic paths, all of the same colour, which cover the entire vertex set. They conjectured that it is possible to replace $2\sqrt{n}$ by $\sqrt{n}$. We prove this to be true for all sufficiently large $n$. |
| title | A proof of a conjecture of Erdős and Gyárfás on monochromatic path covers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.03623 |