Definable K\H{o}nig theorems

TL;DR

This paper establishes bounds on the Baire measurable and -measurable edge chromatic numbers of certain Borel graphs on Polish spaces, extending classical Kf6nig theorems to a definable setting.

Contribution

It proves that Baire measurable edge chromatic number is at most (G)+1 for graphs with no odd cycles and maximum degree (G), extending Kf6nig's theorem to a definable context.

Findings

Baire measurable edge chromatic number a3 (G)+1 for graphs with no odd cycles.

a3 -measurable edge chromatic number bound for b5-hyperfinite graphs.

Edge chromatic number bounded by maximum degree plus asymptotic separation index.

Abstract

Let $X$ be a Polish space with Borel probability measure $\mu,$ and let $G$ be a Borel graph on $X$ with no odd cycles and maximum degree $\Delta(G).$ We show that the Baire measurable edge chromatic number of $G$ is at most $\Delta(G)+1$, and if $G$ is $\mu$-hyperfinite then the $\mu$-measurable edge chromatic number obeys the same bound. More generally, we show that $G$ has Borel edge chromatic number at most $\Delta(G)$ plus its asymptotic separation index.