Overfullness of edge-critical graphs with small minimal core degree

Abstract

Let $G$ be a simple graph. Let $\Delta \left(G\right)$ and $\chi \prime \left(G\right)$ be the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $\mid E\left(G\right)\mid ∕\lfloor \mid V\left(G\right)\mid ∕2\rfloor >\Delta \left(G\right)$, and critical if $\chi \prime \left(H\right)<\chi \prime \left(G\right)$ for every proper subgraph $H$ of $G$. Clearly, if $G$ is overfull then $\chi \prime \left(G\right)=\Delta \left(G\right)+1$. The core of $G$, denoted by $G_{\Delta }$, is the 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 $k\ge 2$, if $G$ is critical with $\Delta \left(G\right)\ge \frac{2}{3}n+\frac{3k}{2}$ and $\delta \left(G_{\Delta }\right)\le k$, then $G$ is overfull.