K\H{o}nig's Line Coloring and Vizing's Theorems for Graphings

TL;DR

This paper extends classical graph coloring theorems to the setting of graphings, showing measurable edge-colorings with near-optimal color bounds and conditions for achieving Vizing's bound.

Contribution

It proves the existence of measurable edge-colorings for graphings with bounds close to Vizing's theorem, including special cases with no odd cycles.

Findings

Every graphing of maximum degree d admits a measurable edge-coloring with d + O(√d) colors.

If the graphing has no odd cycles, then d+1 colors suffice.

A conjecture about finite graphs could imply that d+1 colors are always enough.

Abstract

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 K\H{o}nig, $d$ 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, orbit equivalence theory and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge-coloring with $d + O(\sqrt{d})$ 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.