Edge coloring graphs with large minimum degree

Let G be a simple graph with maximum degree Delta(G). A subgraph H of G is overfull if.[E (H)] >Delta(G).[V (H)]/2.. Chetwynd and Hilton in 1986 conjectured that a graph G with Delta(G) >IV (G)1/3 has chromatic index Delta(G) if and only if G contains no overfull subgraph. The best previous results supporting this conjecture have been obtained for regular graphs. For example, Perkovic and Reed verified the conjecture for large regular graphs G with degree arbitrarily close to IV (G). /2. We provide a similar result for general graphs asymptotically, showing 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 >= n(0) vertices with minimum degree at least (1 +epsilon) n, then G has chromatic index Delta(G) if and only if G contains no overfull subgraph.