Overfullness of edge-critical graphs with small minimal core degree

Let G be a simple graph. Let G Delta(G) and chi G '() be the maximum degree and the chromatic index of G, respectively. We call G overfull if EGVG()() 2//G>Delta(), andcriticalif chi H chi G '()<'()for every propersubgraphHofG. Clearly, ifGis overfull then chi G G '()=Delta()+1. Thecore of G, denoted byG Delta,isthe subgraph of G induced by all its maximum degree vertices. We believe that utilizing the core degree condition could be considered as an approach to attack the overfull conjecture. Along this direction, we in this paper show that for any integer k2 >=, if G is critical with G n Delta()+k2332 >= and delta Gk()Delta <=, then G is overfull.