  Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of
$G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and
Hilton in 1985 conjectured that a graph $G$ with $\Delta(G)>|V(G)|/3$ has
chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph.
The 1-factorization conjecture is a special case of this overfull conjecture,
which states that for even $n$, every regular $n$-vertex graph with degree at
least about $n/2$ has a 1-factorization and was confirmed for large graphs in
2014. Supporting the overfull conjecture as well as generalizing the
1-factorization conjecture in an asymptotic way, in this paper, we show that
for any given $0<\varepsilon <1$, there exists a positive integer $n_0$ such
that the following statement holds: if $G$ is a graph on $2n\ge n_0$ vertices
with minimum degree at least $(1+\varepsilon)n$, then $G$ has chromatic index
$\Delta(G)$ if and only if $G$ contains no overfull subgraph.

Combinatorics