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.
If we consider only regular graphs, which one of them are bigger in the class of regular graphs (contains more graphs), and why?
In a series of four papers recently posted on the arXiv, Béla Csaba, Daniela Kühn, Allan Lo, Deryk Osthus and Andrew Treglown show that every large $D$-regular even graph with $D \geq 2\lceil n/4\rceil -1$ can be covered by edge-disjoint matchings, i.e. is class 1.
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$:
(i) [1-factorization conjecture] Suppose that $n$ is even and $D \geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $\chi'(G)=D$.
(ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching.
(iii) [Optimal packings of Hamilton cycles] Suppose that $G$ is a graph on $n$ vertices with minimum degree $\delta\ge n/2$. Then $G$ contains at least ${\rm reg}_{\rm even}(n,\delta)/2 \ge (n-2)/8$ edge-disjoint Hamilton cycles. Here ${\rm reg}_{\rm even}(n,\delta)$ denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on $n$ vertices with minimum degree $\delta$. According to Dirac, (i) was first raised in the 1950s. (ii) and the special case $\delta= \lceil n/2 \rceil$ of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible. In the current paper, we prove the above results for the case when $G$ is close to the union of two disjoint cliques.
For (not necessarily regular) graphs of maximum degree down to $n/3$ the overfull conjecture is that the only obstruction to being class 1 is an odd order subgraph $H$ with more than $\Delta(G)(|H|-1)$ edges (too many to $\Delta$-colour).
Added by later editors:
There is also an old result by Richard Stong regarding 1-factorizability of even Cayley graphs:
Stong, Richard A., On 1-factorizability of Cayley graphs, J. Comb. Theory, Ser. B 39, 298-307 (1985). ZBL0581.05031.
If you don't consider regular graphs, but all graphs of a given order, the answer is in the following paper of Frieze, Jackson, McDiarmid and Reed (1986)
http://www.math.cmu.edu/~af1p/Texfiles/EDGECOLOR.pdf
They prove that for any $\epsilon >0$ and for any sufficiently large $n$, the proportion of class 2 graphs among all graphs of order $n$ is less than $n^{-(\tfrac18-\epsilon)n}$.
You can probably use some of their ideas to fix a given degree $d$ and a order $n$ and look at the proportion of class 2 graphs among all graphs of order $n$ and degree $d$.
Robinson and Wormald's famous proof that almost all cubic graphs are hamiltonian settles the cubic case, since a hamiltonian cubic graph is necessarily 3-edge colourable.
