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.

We were informed by Gabor Elek that our result about bounded cocycle (Theorem 1.2) was previously established by Gabriella Kuhn. Reference and discussion of this result has been added