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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.

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Yes. Choose any proper edge coloring with $\Delta+1$ colors (it exists by Vizing theorem). If we have two color classes with $a$ and $b$ edges respectively, $a\geqslant b+2$ (say, $a$ red eges and $b$ blue edges), consider the graph formed by these $a+b$ red or blue edges. It contains a component with more red edges then blue edges. This is a path starting and ending with a red edge. Replace the red edges in this path to blue and vice versa. Now we have $a-1$ red edges and $b+1$ blue edges. Something decreased, for example the sum of squares of the sizes of color classes. Therefore after finitely many operations we come with a situation when the sizes differ at most by 1, that we need.

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  • $\begingroup$ So this reasoning can be extended to any (non-regular) graphs, right $\endgroup$ – vidyarthi Jul 2 '19 at 10:41
  • $\begingroup$ @vidyarthi of course $\endgroup$ – Fedor Petrov Jul 2 '19 at 10:51

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