Kőnig’s Line Coloring and Vizing’s Theorems for Graphings

The classical theorem of Vizing states that every graph of maximum degree d admits an edge-coloring with at most d + 1 colors. Furthermore, as it was earlier shown by Kőnig, d colors suffice if the graph is bipartite. We investigate the existence of measurable edge-colorings for graphings. A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits and orbit equivalence theory. We show that every graphing of maximum degree d admits a measurable edge-coloring with d + O( √ d) colors; furthermore, if the graphing has no odd cycles, then d + 1 colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that d+ 1 colors are always enough.