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KŐNIG’S LINE COLORING AND VIZING’S THEOREMS FOR GRAPHINGS
Published online by Cambridge University Press: 19 September 2016
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 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.
MSC classification
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- Research Article
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- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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- © The Author(s) 2016
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