Measurable versions of Vizing's theorem

Abstract

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively Δ and π. The “approximate” version states that, for any Borel probability measure on the edge set and any ϵ>0, we can properly colour all but ϵ-fraction of edges with Δ+π 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 Δ+π colours.