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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 largegraph 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 faras we know, this is the first result on the conjecture for graphs of odd orderand with a minimum degree constraint.

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