  Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of
$G$ is overfull if $|E(H)|>\Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$.
Chetwynd and Hilton in 1986 conjectured that a graph $G$ with
$\Delta(G)>\frac{1}{3}|V(G)|$ has chromatic index $\Delta(G)$ if and only if
$G$ contains no overfull subgraph. Let $0<\varepsilon <1$ and $G$ be a large
graph on $n$ vertices with minimum degree at least
$\frac{1}{2}(1+\varepsilon)n$. It was shown that the conjecture holds for $G$
if $n$ is even. In this paper, the same result is proved if $n$ is odd. As far
as we know, this is the first result on the conjecture for graphs of odd order
and with a minimum degree constraint.
