# Maximum number of 1-factors in a color class

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?

• If the regular class 2 graph has a perfect matching, one can choose the matching to be one of the color class. – Bullet51 May 2 at 8:27
• @Bullet51 so then, the question reduces to how many perfect matchings/maximum matchings there are in the graph, right? – vidyarthi May 2 at 10:13