Measurable Vizing’s theorem
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 $\Delta \in \mathbb {N}$ defined on a standard probability space $(X,\mu )$ admits a $\mu $ -measur...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000835/type/journal_article |
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| Summary: | 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
$\Delta \in \mathbb {N}$
defined on a standard probability space
$(X,\mu )$
admits a
$\mu $
-measurable proper edge coloring with
$(\Delta +1)$
-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
$\mu $
is
$\mathcal {G}$
-invariant. |
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| ISSN: | 2050-5094 |