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Let $G$ be a simple graph. Denote by $n$, $\Delta(G)$ and $\chi' (G)$ be theorder, the maximum degree and the chromatic index of $G$, respectively. We call$G$  \emph{overfull} if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and {\itcritical} if $\chi'(H) < \chi'(G)$ for every proper subgraph $H$ of $G$.Clearly, if $G$ is overfull then $\chi'(G) = \Delta(G)+1$. The \emph{core} of$G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by all its maximumdegree vertices. We believe that utilizing the core degree condition could beconsidered as an approach to attacking the overfull conjecture. Along thisdirection, we in this paper show that for any integer $k\geq 2$, if $G$ iscritical with $\Delta(G)\geq \frac{2}{3}n+\frac{3k}{2}$ and$\delta(G_\Delta)\leq k$, then $G$ is overfull.

Overfullness of edge-critical graphs with small minimal core degree