Bounded diameter monochromatic component covers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908750005141504 |
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| author | Pokrovskiy, Alexey |
| author_facet | Pokrovskiy, Alexey |
| contents | Ryser conjectured that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees. Motivated by a question of Austin in analysis, Milićević predicted something stronger -- that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees \emph{of bounded diameter}. Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Milićević's Conjecture, most notably that it is true for $r=5$. We also obtain several new cases of a generalization of Milićević's Conjecture to non-complete graphs due to DeBiasio-Kamel-McCourt-Sheats. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05842 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded diameter monochromatic component covers Pokrovskiy, Alexey Combinatorics 05D15 G.2.2 Ryser conjectured that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees. Motivated by a question of Austin in analysis, Milićević predicted something stronger -- that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees \emph{of bounded diameter}. Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Milićević's Conjecture, most notably that it is true for $r=5$. We also obtain several new cases of a generalization of Milićević's Conjecture to non-complete graphs due to DeBiasio-Kamel-McCourt-Sheats. |
| title | Bounded diameter monochromatic component covers |
| topic | Combinatorics 05D15 G.2.2 |
| url | https://arxiv.org/abs/2507.05842 |