The chromatic index of graphs with large even order n and minimum degree at least 2n/3

Abstract

The 1-Factorization Conjecture states that any regular graph of even order n in which the vertex degrees are at least [Formula presented] has a factorization of its edges into perfect matchings. Chetwynd and Hilton [1], [3] and independently Niessen and Volkmann [11] verified this conjecture when the vertex degrees are at least .83n. This is equivalent to saying that any such regular graph G has chromatic index χ′(G) equal to its maximum degree Δ(G). Using that result, it was shown in [14] that for any graph G (not necessarily regular) of even order n and minimum degree at least .882n, χ′(G)=Δ(G) if and only if there is no vertex v for which the number of edges of G−v surpasses [Formula presented]. Recently, Csaba, Kühn, Lo, Osthus and Treglown [6] verified that the 1-Factorization Conjecture holds for all graphs with sufficiently large n. Using this result, we show that for any graph with even order n sufficiently large and minimum degree at least [Formula presented] if and only if no vertex-deleted subgraph has more than [Formula presented] edges. These results are presented in the context of the more general Overfull Conjecture.