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