Measurable Vizing's theorem
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910544127066112 |
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| author | Grebík, Jan |
| author_facet | Grebík, Jan |
| contents | We prove a full measurable version of Vizing's theorem for bounded degree Borel graphs, that is, we show that every Borel graph $\mathcal{G}$ of degree uniformly bounded by $Δ\in \mathbb{N}$ defined on a standard probability space $(X,μ)$ admits a $μ$-measurable proper edge coloring with $(Δ+1)$-many colors. This answers a question of Marks [Question 4.9, J. Amer. Math. Soc. 29 (2016)] also stated in Kechris and Marks as a part of [Problem 6.13, survey (2020)], and extends the result of the author and Pikhurko [Adv. Math. 374, (2020)] who derived the same conclusion under the additional assumption that the measure $μ$ is $\mathcal{G}$-invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_16440 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Measurable Vizing's theorem Grebík, Jan Logic We prove a full measurable version of Vizing's theorem for bounded degree Borel graphs, that is, we show that every Borel graph $\mathcal{G}$ of degree uniformly bounded by $Δ\in \mathbb{N}$ defined on a standard probability space $(X,μ)$ admits a $μ$-measurable proper edge coloring with $(Δ+1)$-many colors. This answers a question of Marks [Question 4.9, J. Amer. Math. Soc. 29 (2016)] also stated in Kechris and Marks as a part of [Problem 6.13, survey (2020)], and extends the result of the author and Pikhurko [Adv. Math. 374, (2020)] who derived the same conclusion under the additional assumption that the measure $μ$ is $\mathcal{G}$-invariant. |
| title | Measurable Vizing's theorem |
| topic | Logic |
| url | https://arxiv.org/abs/2303.16440 |