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arXiv · 1905.01716

Measurable versions of Vizing's theorem

Jan Grebík·Oleg Pikhurko
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Abstract

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $\Delta$ and $\pi$. The ``approximate'' version states that, for any Borel probability measure on the edge set and any $\epsilon>0$, we can properly colour all but $\epsilon $-fraction of edges with $\Delta+\pi$ colours in a Borel way. The ``measurable'' version, which is our main result, states that if, additionally, the measure is invariant, then there is a measurable proper edge colouring of the whole edge set with at most $\Delta+\pi$ colours.

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Jan Grebík, Oleg Pikhurko. 2019-05-05. Measurable versions of Vizing's theorem. https://arxiv.org/abs/1905.01716

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