Edge coloring graphs with large minimum degree

LetG $G$ be a simple graph with maximum degreeΔ ( G )${\rm{\Delta }}(G)$ . A subgraphH $H$ ofG $G$ is overfull if∣ E ( H ) ∣ > Δ ( G ) ⌊ ∣ V ( H ) ∣ ∕ 2 ⌋ . $| E(H)| \gt {\rm{\Delta }}(G)\lfloor | V(H)| \unicode{x02215}2\rfloor .$ Chetwynd and Hilton in 1986 conjectured that a graphG $G$ withΔ ( G ) > ∣ V ( G ) ∣ ∕ 3 ${\rm{\Delta }}(G)\gt | V(G)| \unicode{x02215}3$ has chromatic indexΔ ( G )${\rm{\Delta }}(G)$ if and only ifG $G$ contains no overfull subgraph. The best previous results supporting this conjecture have been obtained for regular graphs. For example, Perković and Reed verified the conjecture for large regular graphsG $G$ with degree arbitrarily close to∣ V ( G ) ∣ ∕ 2 $| V(G)| \unicode{x02215}2$ . We provide a similar result for general graphs asymptotically, showing that for any given0 < ϵ < 1 $0\lt \epsilon \lt 1$ , there exists a positive integern 0${n}_{0}$ such that the following statement holds: ifG $G$ is a graph on2 n ≥ n 0$2n\ge {n}_{0}$ vertices with minimum degree at least( 1 + ϵ ) n $(1+\epsilon )n$ , thenG $G$ has chromatic indexΔ ( G )${\rm{\Delta }}(G)$ if and only ifG $G$ contains no overfull subgraph.