Edge coloring graphs with large minimum degree

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