The overfull conjecture on graphs of odd order and large minimum degree

Let $G$ be a simple graph with maximum degree $\Delta \left(G\right)$. A subgraph $H$ of $G$ is overfull if $\mid E\left(H\right)\mid >\Delta \left(G\right)\lfloor \frac{1}{2}\mid V\left(H\right)\mid \rfloor$. Chetwynd and Hilton in 1986 conjectured that a graph $G$ with $\Delta \left(G\right)>\frac{1}{3}\mid V\left(G\right)\mid$ has chromatic index $\Delta \left(G\right)$ if and only if $G$ contains no overfull subgraph. Let $0<\epsilon <1$, $n$ be sufficiently large, and $G$ be graph on $n$ vertices with minimum degree at least $\frac{1}{2}\left(1+\epsilon \right)n$. 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 Overfull Conjecture for graphs of odd order and with a minimum degree constraint.