Geodesic
KŐNIG’S LINE COLORING AND VIZING’S THEOREMS FOR GRAPHINGS
Forum of Mathematics, Sigma, Tome 4 (2016)

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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 (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence 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. 
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     author = {ENDRE CS\'OKA and G\'ABOR LIPPNER and OLEG PIKHURKO},
     title = {K\H{O}NIG{\textquoteright}S {LINE} {COLORING} {AND} {VIZING{\textquoteright}S} {THEOREMS} {FOR} {GRAPHINGS}},
     journal = {Forum of Mathematics, Sigma},
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ENDRE CSÓKA; GÁBOR LIPPNER; OLEG PIKHURKO. KŐNIG’S LINE COLORING AND VIZING’S THEOREMS FOR GRAPHINGS. Forum of Mathematics, Sigma, Tome 4 (2016). doi: 10.1017/fms.2016.22

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