# Measurable versions of Vizing's theorem

## TL;DR

This paper proves two measurable versions of Vizing's theorem for Borel multi-graphs, providing approximate and exact measurable proper edge colourings under certain boundedness and invariance conditions.

## Contribution

It introduces the first measurable versions of Vizing's theorem for Borel multi-graphs, including an approximate and an invariant measure-based exact colouring result.

## Key findings

- Approximate colouring with 5-fraction of edges coloured using 5+ 5 colours.
- Existence of a measurable proper edge colouring with 5+ 5 colours under invariant measure.
- Main result extends classical Vizing's theorem to a measurable setting for Borel multi-graphs.

## Abstract

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $\Delta$ and $\pi$. The ``approximate'' version states that, for any Borel probability measure on the edge set and any $\epsilon>0$, we can properly colour all but $\epsilon $-fraction of edges with $\Delta+\pi$ colours in a Borel way. The ``measurable'' version, which is our main result, states that if, additionally, the measure is invariant, then there is a measurable proper edge colouring of the whole edge set with at most $\Delta+\pi$ colours.