Edge coloring graphs with large minimum degree

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 $G$ with $\mathrm{\Delta }(G)>\mathrm{\mid }V(G)\mathrm{\mid }\mathrm{/}3$ has chromatic index $\mathrm{\Delta }(G)$ 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 $n\mathrm{/}2$ 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 $0<\epsilon <1$, 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 $(1+\epsilon )n$, then $G$ has chromatic index $\mathrm{\Delta }(G)$ if and only if $G$ contains no overfull subgraph.