Consider any graph with $n$ vertices and maximum degree $\Delta$. By Vizing's theorem, the graph could be edge colored(properly) with at most $\Delta+1$ colors.
My question pertains as to what the maximum and minimum number of 1-factors(independent edges) we could put in a color class. For regular Class $1$ graphs, the answer should be exactly $\frac{n}{2}$ edges. But, what for an arbitrary graph? What if the graph were regular(Class 2)? What if, in addition it were vertex transitive, or, even better, Cayley?
