Chromatic index of dense quasirandom graphs

Let $G$ be a simple graph with maximum degree $\mathrm{\Delta }(G)$. A subgraph $H$ of $G$ is overfull if $\mathrm{\mid }E(H)\mathrm{\mid }>\mathrm{\Delta }(G)\lfloor \mathrm{\mid }V(H)\mathrm{\mid }\mathrm{/}2\rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $\mathrm{\Delta }(G)>n\mathrm{/}3$ has chromatic index $\mathrm{\Delta }(G)$ if and only if $G$ contains no overfull subgraph. Glock, K"{u}hn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, K"{u}hn and Osthus is affirmative.