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Measurable Vizing’s theorem
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AbstractWe 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.
Title: Measurable Vizing’s theorem
Description:
AbstractWe 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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