Measurable Vizing’s theorem

Abstract

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.

Keywords

• 03E15
• 60C05
• 05C15
• 37A50