Consider a simple regular graph $G$ with $n$ vertices and $E$ edges. Then, can we say that the edges can be colored equitably in $\Delta+1$ colors? By equitability is meant that a proper $\Delta+1$ coloring has either $\left\lceil\frac{E}{\Delta+1}\right\rceil$ or $\left\lfloor\frac{E}{\Delta+1}\right\rfloor$ independent edges in each color class.

I think yes. By the way, an equitable $\Delta+1$ coloring of vertices exists by Hajnál-Szemerédi theorem. I think the same would apply for edge coloring too. If not, can we at least say that each color class would have at least $\left\lfloor\frac{E}{\Delta+1}\right\rfloor$ independent edges? Thanks beforehand.