class 1 vs class 2 of graphs Vizing's theorem states that a graph can be edge-colored in either $\Delta$ or $\Delta+1$ colors, where $\Delta$ is the maximum degree of the graph.
A graph with edge chromatic number equal to $\Delta$ is known as a class 1 graph.
A graph with edge chromatic number equal to $\Delta+1$ is known as a class 2 graph.
which one of them are bigger?(contains more graphs) and why?
 A: Well, it really depends what you mean by bigger. 
As it is asked, the answer is: they are both the same size.
Since every graph can be realized up to isomorphism as a graph on $V=\{1,2,3,.., n\}$ and because for each fixed $n$ the set of graphs with $V=\{1,2,3,.., n\}$ is finite (or countable if you include multiedges/loops) it follows that up to isomorphism there are countable many graphs.
Now the Class 1 contains the countably many graphs $K_{2n}$, while Class 2 the countably many graphs $K_{2n+1}$, thus both Classes are countable. This proves that you can construct a bijection from Class 1 to Class 2.
If you don't like the up to isomorphism part, there is nothing you can do, as all the graphs don't form a set.
But there are other ways in which we can define what bigger means. For example, Erdos and Wilson proved that if for each $n$ you count all graphs on $\{1,2,3,.., n\}$ of Class 1 and Class 2, the ratio those two numbers approaches 1 when $n \to \infty$. So, in this sense, almost all graphs are of Class 1.
