Measurable local Vizing theorem

Establish a measurable version of local Vizing’s theorem: for every measured graph, construct a measurable edge coloring with Δ+1 colors such that each edge e={v,w} receives a color at most max{deg(v),deg(w)}+1.

Background

Christiansen proved a local version of Vizing’s theorem for finite graphs. The paper states that this argument does not appear to generalize automatically to the measurable setting.